Stochastic Simulation of Hypothetical MH370 End-of-Flight Scenarios
Brock McEwen
May 15, 2015
Executive Summary
This paper stochastically models MH370 end-of flight dynamics, and finds no evidence to support last month’s
announced continuation of status quo search strategies.
This finding is consistent with a documented pattern of decisions by MH370 search leaders which make no sense. A
rigorous, independent audit of both the Inmarsat data’s entire chain of custody and MH370 search leadership is
recommended.
Introduction
In a joint statement issued April 16, 2015, Malaysian Transport Minister Liow Tiong Lai, Australian Deputy Prime
Minister Warren Truss and Chinese Transport Minister Yang Chuantang indicated the search for Malaysian Airlines
Flight 370 (MH370) would continue to focus on the deep-sea search in the Southern Indian Ocean (SIO):
“Should the aircraft not be found within the current search area, ministers agreed to extend the search by an
additional 60,000 square kilometres to bring the search area to 120,000 square kilometres and thereby cover
the entire highest probability area identified by expert analysis,” they said in a joint statement.
“Ministers recognise the additional search area may take up to a year to complete given the adverse weather
conditions in the upcoming winter months.”
In committing to this extension – described graphically as a modest search zone expansion in all four directions search leaders implicitly assume that
1) the Inmarsat data is sufficiently accurate and precise to permit interpretation by investigators AND
2) the conclusion they drew (flight to fuel exhaustion, followed by pilotless spiral to impact) is correct, YET
3) the wreckage this conclusion predicts remains outside the areas already searched by side-scan
This paper tests this hypothesis, via stochastic simulation of plausible post-fuel exhaustion flight paths.
Stochastic (or “Monte Carlo”) simulation
Stochastic (“involving random numbers”) simulation (“running of many, many trials”) is a well-recognized means of
analyzing complex systems – in particular, whenever one or more of a system’s inputs is unknown (or “known” to vary
within a range). Simulation involves building a model which treats the input as constant, but assigns it a randomly
generated value (governed by distributional parameters which define the limits of plausibility). The resulting model
(inputs and outputs) are referred to as a single “trial”. The model is then run again, with a different randomly
generated plausible value – and so on.
The greater the number and complexity of variables driving a system, the more valuable a tool simulation becomes.
The key output is no longer a single “best estimate” or “maximum likelihood” answer – rather, a distribution of results,
from which statistical inferences can be drawn.
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Model starting point: second engine fuel exhaustion
The compatibility of assumption 3) with 1) and 2) will be tested by assuming 1) and 2) are true, and developing the a
priori probability of 3). We therefore begin the simulation from the point of fuel exhaustion for MH370’s second of
two engines, as assumed under 1) and 2). This starting point (in time and space) is appropriate, as the flight path up to
this point is relatively stable, while the flight path after this point will be relatively erratic.
Consider the assumed state of MH370 at second engine fuel exhaustion to be given by four attributes:
a)
b)
c)
d)
bearing
speed
altitude
distance to 7th arc (FL0) in the direction of a)
a) 186 degrees is selected as a baseline, being representative of predictions conforming to 1) and 2).
b) While consensus under 2) is strong that a final major turn did send MH370 into the SIO, its timing (within a 12
minute span) remains in dispute. Accordingly, a range (albeit narrow) of assumed pre-exhaustion speeds is possible.
We choose 480 KGS as a representative baseline speed, for consistency with parallel analysis.
c) FL350 is chosen as a baseline altitude; this is the consensus assumption under 1) and 2), and flight simulation
experience conducted by Mike Exner, and reported by Brian Anderson - both members of the MH370 Independent
Grouop (or “IG”) - suggest a 777 essentially maintains altitude for the first 4-5 minutes on one engine.
d) 21 nmi is selected as the forced result, under 1) and 2), of
i.
ii.
iii.
iv.
our selections of a), b) and c)
per Exner/Anderson, flight simulator result of a constant deceleration of 0.315 kts/sec while on one engine
engine 1 fuel exhaustion time: deemed to have been signalled by the data packet sent at exactly 00:11:00
engine 2 fuel exhaustion time: assumed baseline of 00:15:49.42, based on the following reasoning:
a) signal data packet sent at 00:19:29.42 (7th arc) has been deemed to represent a log-on request
b) trigger for this event is deemed to be engine two fuel exhaustion, causing total power loss
c) time from power loss to log-on request has been claimed to be 3:40 (i.e. 220 seconds, +/-10)
By subtraction, the point of engine two fuel exhaustion is 00:15:49.42, +/- 10 seconds. 21 nmi is then simply 56 nmi
(distance from first engine exhaustion to 7th arc (FL0) minus the 35 nmi MH370 would travel in time t=289.5s starting
at velocity v=0.1333nmi/s, with acceleration a=-0.0000875 nmi/s2 (d = vt + ½at2 = 35 nmi)
ALL of the above baseline assumptions will be sensitivity-tested.
The frame of reference is shifted from geographic (putting MH370 at roughly [S38, E89] and bearing = 186 degrees) to
a Cartesian frame (x and y in nmi, z = feet) with initial position [0, 0, 35000], and bearing 180.
Model time-step: 10 seconds
The model recalculates all flight dynamics every 10 seconds. This value was chosen to strike a reasonable balance
between the precision of each scenario, and model simplicity sufficient to permit generation of a large number of
scenarios. In the notation that follows, the time index in parentheses represents units of ten seconds.
Number of simulated paths: 100,000
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Model Inputs: Flight Dynamics after Second Engine Fuel Exhaustion
Note: “U(a, b)” = random variable with values uniformly distributed between a and b

Deceleration: a stochastic process was built, which begins at (and whose mean value is) -0.315, but then
randomly walks, based on the formula:
a(0) = -0.315 KGS/s; a(t+1) = a(t) + U(-0.05, 0.05)

Turn rate: bearing, b, follows a left-biased random walk, with exponentially increasing (but capped) bias, 𝑙:
b(0) = 180b(t+1) = b(t) – min[𝑙(t)√t -1, cap(t)]
cap(t) = tan(10g/2.5v(t))

y(0)=0 nmi; y(t+1) = y(t) + 10 / 3600 * v(t) * cos(b(t))
Vertical velocity, i, starts at -250’/min, and follows a complex random walk gravitating towards -10,000’/min:
i(0) = -250’/min;
j(0) = -3600’/min;

𝑙(0) = 1.15; 𝑙(t+1) = 𝑙(t) + U(-0.027, 0.033)
Horizontal position: is derived in straightforward fashion:
x(0)=0 nmi; x(t+1) = x(t) + 10 / 3600 * v(t) * sin(b(t))

v(0) = 388.8KGS; v(t+1) = max[100, v(t) + 10 * a(t)]
i(t+1) = min[0, i(t) + 25%(j(t) - i(t)) + U(-20, 20)
j(t+1) = j(t) + 12.5%(10000’/min – j(t)) + U(-45, 45) + 0.5b(t)
Altitude: is derived deterministically from prior vertical position, plus effect of vertical velocity, i:
z(0) = 35,000’; z(t+1) = z(t) + 10 * i(t)
The left bias reflects a consensus view under 2) that a left turn developed at t=0. While the model permits coming out
of spirals - and even right turns - such events are infrequent, as the bias tends to produce increasingly tight left turns.
The cap on the change in bearing per second reflects basic principles of flight: the faster a plane is traveling, the harder
it is to turn quickly. A basic relationship was developed, and calibrated to Exner-reported flight simulator results, with
respect to both terminal radius distributions (2-15 nmi) and maximum bank angle (1/sec).
While the vertical velocity formula may seem complex, a two-factor model was required to control vertical speed AND
acceleration. Having vertical speed “chase” a target of j - which itself is “chasing” a target of 10,000’/minute - not only
achieves this, but generates a much richer array of descent patterns than could be achieved by a one-factor model.
The last term in the “j” formula introduces a modest correlation between bank angle, b, and vertical velocity. This
additional term simply reflects the fact that altitude becomes more difficult to maintain as bank angle increases.
Interestingly, even though the turn radius is capped at 1.5 /second (to reflect the limits of large aircraft dynamics),
typical spirals still tended to exhibit progressively smaller turn radii, due to decreasing velocity.
Model Output: Distance between Impact and 7th Arc
The trial ends when altitude reaches zero – at which point, two positions are recorded:
1) coordinates (and other metrics) at 00:19:29
2) coordinates (and other metrics) at time of impact
The distance between 1) and the 7th arc (FL(z)) is then used to determine whether that trial “hit” the 7th arc. The
tolerance for defining “hit” was set at 0.3 nmi in the y direction (or roughly 0.22 nmi crow’s flight).
For “hits”, the distance between 2) and the 7th arc (FL0) is computed: this is the value whose distribution we seek.
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Results - Summary
-
-
10.2% of paths “hit” the 7th arc
hits tended to do so at relatively high altitudes
o spirals tend to develop gradually, so most paths were relatively straight over the first 2 minutes
o straighter paths tended to overfly the 7th arc (FL0), and thus only hit at higher altitudes
of the paths hitting the 7th arc:
o 62% impact OUTSIDE the 7th arc (FL0)
o 99% impact within the area already searched (from 11nmi inside to 27 nmi outside 7th arc (FL0)
o 0.9% impact OUTSIDE the searched area
o 0.1% impact INSIDE the searched area
Results – Detailed
9000
Distribution GROUND SPEED (in knots) at Impact
8000
7000
6000
5000
4000
Missed 7th Arc
3000
2000
1000
Hit 7th Arc
550-560
500-510
490-500
460-470
450-460
440-450
430-440
420-430
410-420
400-410
370-380
390-400
360-370
-6000--5500
380-390
350-360
-6500--6000
340-350
-7000--6500
330-340
320-330
310-320
300-310
290-300
280-290
270-280
260-270
250-260
240-250
230-240
220-230
210-220
200-210
190-200
180-190
170-180
160-170
150-160
140-150
130-140
120-130
110-120
100-110
0
9000
Distribution of VERTICAL SPEED (in ft/min) at Impact
8000
Calibration reference: Anderson/Exner
flight simulator report (“descent rate
would be up to 15000 feet/minute”)
7000
6000
5000
4000
Missed 7th Arc
3000
2000
1000
Hit 7th Arc
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MH370 End of Flight Modeling v2
-500-0
-1000--500
-1500--1000
-2000--1500
-2500--2000
-3000--2500
-3500--3000
-4000--3500
-4500--4000
-5000--4500
-5500--5000
-7500--7000
-8000--7500
-8500--8000
-9000--8500
-9500--9000
-10000--9500
-10500--10000
-11000--10500
-11500--11000
-12000--11500
-12500--12000
-13000--12500
-13500--13000
-14000--13500
-14500--14000
-15000--14500
-15500--15000
-16000--15500
-16500--16000
-17000--16500
-17500--17000
<-18000
-18000--17500
0
Page 4 of 10
6000
Distribution of TIME from exhaustion to impact (min)
5000
Calibration reference: Exner flight simulator
report (“between 4 and 13 minutes”)
4000
3000
Missed 7th Arc
2000
1000
Hit 7th Arc
0
-4
-6
-8
2
4
6
8
10
12
14
13.0
12.8
12.7
12.5
12.3
12.2
12.0
11.8
11.7
11.5
11.3
11.2
11.0
10.8
10.7
10.5
10.3
10.2
9.8
10.0
9.7
9.5
9.3
9.2
9.0
8.8
8.7
8.5
8.3
8.2
8.0
7.8
7.7
7.5
7.3
7.2
7.0
6.8
6.7
6.5
6.3
6.2
6.0
5.8
5.7
5.5
5.3
5.2
5.0
4.8
4.7
4.5
4.3
4.2
4.0
3.8
3.7
3.5
0
Scatter plot: position at impact
20
Scatter plot:
position at
00:19:29
15
10
5
-10
0
-20 -15 -10 -5 0
-5
-12
-14
-16
All
Hits
10 15 20 25 30 35 40 45 50 55
-10
All
-18
Hits
-20
5
-15
-20
-25
-22
-30
7th arc (FL0)
-35
-24
-40
Comment: the scatter plot of “hits” at 00:19:29 (above left) do not appear to follow the shape of any particular 7th arc
(which, if plotted, would appear as a straight diagonal line). This is because the model does not constrain the altitude
at which a hit could potentially occur: as a result, relatively straight paths will tend to intersect the 7th arc at higher
altitudes, while relatively curved paths will tend to intersect at lower altitudes. Since the arc moves NW as altitude
decreases, the curved pattern emerges. This relationship is further illustrated by examining the relationship between
altitude and time to impact:
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Distribution of ALTITUDE (in ft) vs.LEFTWARD DISPLACEMENT (in nmi) at 00:19:29
70
60
Hit 7th Arc ONLY
50
40
30
32000-33000
29000-30000
26000-27000
23000-24000
20000-21000
17000-18000
20
10
12.25-12.5
2000-3000
12.5-12.75
11.75-12
12-12.25
11.25-11.5
11.5-11.75
10.75-11
11-11.25
10.25-10.5
5000-6000
10.5-10.75
9.75-10
10-10.25
9.25-9.5
9.5-9.75
8.75-9
9-9.25
8.25-8.5
8000-9000
8.5-8.75
7.75-8
8-8.25
7.25-7.5
7.5-7.75
6.75-7
7-7.25
6.25-6.5
11000-12000
6.5-6.75
5.75-6
6-6.25
5.25-5.5
14000-15000
5.5-5.75
<4
4-4.25
4.25-4.5
4.5-4.75
4.75-5
5-5.25
0
Despite the complex turn function formulation, the output – expressed here as a distribution of total number of
revolutions (360turns) from fuel exhaustion until impact – helps develop an intuition for the plausible range of spirals
modelled:
12000
Distribution of REVOLUTIONS between exhaustion and impact
Calibration reference: Exner flight simulator
report (“between 1 and 3 revolutions”)
10000
8000
6000
4000
Average rotation among “hits”: 370
Missed 7th Arc
2000
Hit 7th Arc
MH370 End of Flight Modeling v2
4
3.9
3.8
3.7
3.6
3.5
3.4
3.3
3.2
3.1
3
2.9
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
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0.1
-0.1
0
Page 6 of 10
12000
Distribution of DISTANCE from impact point to 7th arc (FL0), in nmi
10000
8000
6000
Already searched:
from 12 nmi inside
to 27 nmi outside
of 7th arc (FL0)
4000
Missed 7th Arc
2000
Hit 7th Arc
<-32
-31--30
-30--29
-25--24
-24--23
-23--22
-22--21
-21--20
-20--19
-19--18
-18--17
-17--16
-16--15
-15--14
-14--13
-13--12
-12--11
-11--10
-10--9
-9--8
-8--7
-7--6
-6--5
-5--4
-4--3
-3--2
-2--1
-1-0
0-1
1-2
2-3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
22-23
23-24
24-25
25-26
26-27
27-28
28-29
29-30
30-31
31-32
32-33
33-34
34-35
35-36
36-37
37-38
38-39
39-40
40-41
41-42
42-43
45-46
46-47
47-48
0
Observations:
-
the area searched appears to align reasonably well with the modelled distribution of distances
-
the search to date thus appears to test (and, in fact, to emphatically reject) the hypothesis efficiently
-
while some regions of the 7th arc FL(0) have not been searched out to 12 nmi inside, such “skipped” areas are
over 40nmi to the east of the easternmost of the IG’s newly revised best-estimate path predictions – the
model suggests no material probability of spirals being able to stretch that far to the east.
Sensitivity Testing
Scenario
Baseline
Vertical acceleration +25%
(-0.315-0.394)
Initial altitude +5,000’
(FL350FL400)
Initial velocity reduced 20 knots
(480460 KGS)
Logon sequence takes less time
(3:403:30)
Left-turn bias decreased
1
(𝑙(0), U each  3 x baseline)
Left-turn bias increased
(𝑙(0), U each  2 x baseline)
Arc 6 south of indicated position
(2119 nmi)
Arc 6 north of indicated position
(2123 nmi)
Impact relative to area searched*
Inside
Within
Outside
0.1%
99.0%
0.9%
Comment
as above
Reduced minutes aloft, which reduces
range
Increased minutes aloft, offset by drift
south in 7th arc intersections
Origin moves 1.6nmi further from 7th arc
and speeds drop (doubly hard to reach)
Origin moves ~1nmi closer to 7th arc and
initial speed reduces ~3 KGS
OUTSIDE probability increases, but
average total rotation drops to 207
Paths curl too tightly to make it to 7th
arc; only 8 of 100,000 trials “hit”
0.0%
99.6%
0.4%
0.1%
98.8%
1.1%
0.0%
100.0%
0.0%
0.1%
99.2%
0.7%
0.1%
95.2%
4.6%
0.0%
100.0%
0.0%
0.1%
99.2%
0.8%
Origin moves 2nmi closer to 7th arc
0.1%
99.5%
0.5%
Origin moves 2nmi further from 7th arc
* hits only; “area searched” is defined as a range extending from 12nmi inside to 27nmi outside of the 7th arc (FL0)
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Conclusions
1) Model results point to a stark conclusion: if both the Inmarsat signal data and its interpretation by search
officials is valid, then the search should have turned up wreckage by now (98% probability). This offers strong
circumstantial evidence that either the Inmarsat data or its interpretation is invalid.
2) Accordingly - unless search officials know something we don’t - the announced decision to spend another year
searching around the improbable edges of a discredited theory (while this paper does not address surface
debris, its absence likewise serves as strong Bayesian counter-evidence) is an extremely poor one.
3) Yet another poor decision would be entirely consistent with the string of spectacularly poor decisions search
leaders have made since March, 2014. A careful comparison of these decisions to the actual DATA on which
they have been based is consistent, frankly, with a search which is not being conducted in good faith.
Recommendations
1) A rigorous, fully independent audit of the entire chain of custody of the Inmarsat data should be conducted,
to determine whether any data have been altered, by accident or design, and either before, during, or after its
synthesis and dissemination by Inmarsat.
2) A rigorous, fully independent audit of all key search decisions since March, 2014 should be conducted, to
determine with certainty whether blunders have been due to incompetence or malfeasance. Such an audit
should include all agencies which at any time reported either directly or indirectly to the MH370 Joint
Investigation Team (JIT).
3) Each audit should be conducted under the direct supervision of a government with a) superior experience and
expertise in air accident investigations, b) a strong record of impartiality, and b) no representation on the JIT.
Sources
Anderson, Brian: “The Last 15 minutes of Flight of MH370”. MH370 Independent Group, April 2015
Exner, Mike: personal communications, as well as contributions published to jeffwise.net, circa November, 2014
Acknowledgements
The author wishes to thank the members of the MH370 Independent Group for their expertise, flight simulator hours,
and careful analysis, whose interpretation by the author underpins this model’s calibration.
The author also wishes to express his appreciation for Duncan Steel, Jeff Wise, and others for their skilled moderation
of online forums in which dedicated groups of people from all over the planet continue to crowdsource the unravelling
of one of modern history’s greatest mysteries.
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Appendix: sample stochastically-generated flight paths
(origin = point of second engine exhaustion; down = pre-exhaustion bearing)
0
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
-35
-35
-40
40000
Altitude (ft)
by minute
30000
20000
400
Velocity (KGS)
by minute
300
200
5
10
-40
40000
15
Altitude (ft)
by minute
30000
20000
20
25
400
200
100
10000
100
0
0
0
0
35
40
Velocity (KGS)
by minute
300
10000
0
30
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
-35
-35
-40
40000
30000
20000
Altitude (ft)
by minute
400
300
200
Velocity (KGS)
by minute
5
-40
40000
30000
20000
10
15
Altitude (ft)
by minute
20
25
400
300
200
10000
100
10000
100
0
0
0
0
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30
35
40
Velocity (KGS)
by minute
Page 9 of 10
Appendix: sample stochastically-generated flight paths (cont’d)
(origin = point of second engine exhaustion; down = pre-exhaustion bearing)
0
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
-35
-35
-40
40000
Altitude (ft)
by minute
30000
20000
400
Velocity (KGS)
by minute
300
200
5
10
-40
40000
15
Altitude (ft)
by minute
30000
20000
20
25
400
200
100
10000
100
0
0
0
0
35
40
Velocity (KGS)
by minute
300
10000
0
30
0
0
5
10
15
20
25
30
35
40
0
-5
-5
-10
-10
-15
-15
-20
-20
-25
-25
-30
-30
-35
-35
-40
40000
30000
20000
Altitude (ft)
by minute
400
300
200
Velocity (KGS)
by minute
5
-40
40000
30000
20000
10
15
Altitude (ft)
by minute
20
25
400
300
200
10000
100
10000
100
0
0
0
0
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30
35
40
Velocity (KGS)
by minute
Page 10 of 10