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NOTE: Classification markings (classification banners and portion markings) are redacted
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(Rev. 05- 01-2008)
FEDERAL BUREAU OF INVESTIGATION
Precedence: ROUTINE Date: 04/05/2012
To: Counterterrorism Attn:
San Diego Attn:
New York Attn:
From: San Diego
Contact: SA
Approved By:
Drafted By:
Case ID #:
Title:
MIHDAR MOHAMMAD AL MIHDAR ZAID
aka MOHDAR ABDULLAH
OPERATION ENCORE
OMAR AHMED AL -BAYOUMI
Synopsis: To document the analysis
of a handwritten equation
on a document seized at the residence of OMAR AHMED AL-BAYOUMI in
the United Kingdom (UK). A copy of the document is attached and made
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EO14040-003659
(Rev. 05- 01-2008)
FEDERAL BUREAU OF INVESTIGATION
a part hereto.
Reference:
Attachment(s): A copy of the document seized at the residence
of OMAR AHMED AL-BAYOUMI in the United Kingdom (UK) has been attached
and made a part hereto.
Details: In late
September 2001, a search was conducted by the
Metropolitan Police Service (MPS) in the United Kingdom at the
residence of OMAR AL- BAYOUMI (BAYOUMI). Pursuant to that search,
a document (hereinafter referred to as the document) was seized which
contained a handwritten equation along with other handwritten
calculations and a sketch of what appears to be a plane. On
02/02/2012, Special Agent (SA) and SA
were tasked with analyzing the equation in the document,
which is shown below (Equation 1) as it was written in the document:
Equation (Eq) 1
SA has a Bachelor of Science Degree in Mechanical
Engineering from the University of Utah and SA has a
Bachelor of Science Degree in Aerospace Engineering from Virginia
Tech and a Master of Engineering Degree in Mechanical Engineering
from the University of Connecticut.
Eq 1 above is written exactly as it was written on the document
(without the closing parentheses). Directly below the equation,
there appears to be a sketch of a plane with a vertical dashed line
below it. Next to the dashed line directly below the sketch it was
written that " = hight[sic] the plane from the earth in mile”. At
the bottom of the dashed vertical line, a horizontal line was drawn
forming two sides of a triangle. Next to the horizontal line was
written below which was written “distance from the plane to
hurrizen[sic]”. Furthermore, the equation was taken another step
by squaring both sides and assigning two different values to 2.
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EO14040-003660
The initial analysis of the equation by SA and SA
was that it appeared to be a variation of the Pythagorean
Theorem ( 2 + 2 = c2), an equation used to calculate the length of a
side of a right-triangle when the lengths of two of the sides are
known, see below:
Eq 2: 2 + 2 = 2 Pythagorean Theorem
The diagram below shows a right triangle with each side labeled
, , and from Eq 2:
Diagram 1:
Assuming that the equation is derived from the Pythagorean
Theorem, and given that the units were in miles, as designated by
the writer of the equation, SA and SA concluded that
the numeric constant of 8000 in Eq 1 was most likely referring to
the diameter of the earth (FBI note: the diameter of the earth is
approximately 7,926 miles (About.com)).
Given this assumption, and the fact that the writer of the
equation had provided the object as a “plane” a certain distance above
the earth (
) and a certain distance from the horizon ( ), SA
performed open source checks on a ‘distance to horizon’ equation.
A paper published in 2010 by with the San Diego State
University Astronomy Department used the same type of equation, which
was derived from the Pythagorean Theorem, to calculate the distance
of an object above the earth’s surface to the horizon. Below is a
diagram showing how the calculation is set up:
P
Earth
G
Diagram 2:
C
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EO14040-003661
Points P, C and G (the vertices of the triangle) represent the
“plane” (object above the earth’s surface), the center of the earth,
and a point on the horizon, respectively. Lines
, , and
represent the height of the “plane” (P) from the earth, the distance
from the plane (P) to a point on the horizon (G), and the radius of
the earth ( ), respectively. Line is tangent to the surface of the
earth, making lines and perpendicular at point G. Thus the
triangle shown in Diagram 2 above is a right triangle labeled as
follows:
Diagram 3: ( + )
Below is the Pythagorean Theorem based on the coefficients from
Diagram 3 ( , , and ( + )) which were substituted for , , and
in Eq 2, respectively:
Eq 3 2 + = ( + )2
Note that the value of the hypotenuse (the leg of the triangle
opposite the right angle), which in this case is the radius of the
earth plus the height of the “plane”, is substituted for in equation
. At this point, Eq 3 can be derived to the exact equation
that was written in the document (see Eq 1):
Eq 3 => 2 + = ( + )2
=> 2 + = ( + )( + )
=> 2 + = 2 + + 2
=> = 2 + + 2 - 2
=> = 1 + 2
=> => ≈4000 miles (radius of earth)
=>
EO14040-003662
Eq 4 => => take the square- root of both sides
=>
=> see Eq 1
The writer of the equation on the document plugged in two
different values of into Eq 4. The following two equations (Eq
5 and 6) were also hand- written on the document:
Eq 5
Eq 6
The author of the document plugged values of 2500 and 5000 into
of Eq 4 in order to get Eq 5 and Eq 6 above. These values (2500
and 5000) yield a distance from the "plane" to the point on the horizon
(d) of 50 miles and 70.71 miles, respectively. It appears that the
writer of the equation was attempting to calculate the height (h),
or altitude, of the “plane” when the horizon (G) is 50 miles away
and approximately 70 miles away. It is believed that the purpose
of the equation is to provide the plane’s altitude once a landmark
on the ground appears on the horizon, 50 to 70 miles away. This would
allow for a straight, constant, descending flight-path from that
altitude to the horizon without
any major deviations. According to
, the equation is a rough calculation, given that it
neglects the complexities of terrestrial refraction. Terrestrial
refraction occurs due to the density of the air, which varies with
elevation. This variation in air density affects how light rays
move through the air, which can affect how one observes something
at a distancei.
The author of the document did not calculate the value of
on
the document. Eq 5 and Eq 6 were solved for h by SA using
the values of d2 plugged in by the author of the document, see Eq 5
and 6 below:
Eq 5: => solve for
= 0.312 miles or 1,653 feet
Eq 6: => solve for
= 0.625 miles or 3,300 feet
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EO14040-003663
In addition to the equations and the definitions of the
coefficients on the document, the following calculations were also
written on the document:
Exhibit (Ex) 1: 52 44 44
8 - 25 25
44 80 80
76 76
225 20
20 245
245
The numbers in Ex 1 are basic addition and subtraction
calculations. At this point, agents have been unable to connect
these calculations to the written equations on the document.
However, agents have a theory about what the numbers may represent
based on a few assumptions. First, it is evident that the units of
the equation in the document are miles per the author of the document
whose intent of the equation seemed to be to calculate the height
of a "plane". Second, given that the document was seized at the
residence of BAYOUMI, and given his association with the hijackers
of American Airlines (AA) Flight 77 from Dulles International Airport
(Dulles) to Los Angeles International Airport (LAX), agents assumed
that the numbers may be linked to Dulles and/or flight paths
associated with Dulles and Flight 77.
Based on those assumptions, agents looked at mileages
associated with the flight path of AA Flight 77. Ex 1 shows that
the number 20 was added to the number 225 after the fact. Agents
found that the distance in nautical miles (nm) from Dulles to Reagan
National Airport (Reagan) is approximately 20 nm. If in fact Ex 1
was a calculation of the distance in miles of AA Flight 77 from Dulles
when the takeover occurred, it would make sense that 20 miles would
be added to 225 in order to get a total mileage from the location
of the plane to the Pentagon which is in close proximity to Reagan
National Airport (Reagan).
It is unclear if, or why, BAYOUMI would want to calculate the
mileage from Dulles at the time of the takeover. This information
may have helped provide the hijackers with a time/location that
coordinated with the takeover of other planes on 9/11, or possibly
provided a signal for all five hijackers on Flight 77 to prepare to
take the plane. If this is the case, then there would have had to
have been some sort of landmark on the ground that would have signaled
that the mileage had been met, i.e. a river, mountain range, lake,
etc.
EO14040-003664
EO14040-003665
To: Counterterrorism From: San Diego
Re: 04/05/2012
is a pilot who flew for the United States (U.S.) Navy for
20 years and commercially for American Airlines (AA) for 15 years.
He retired from AA in July of 2001. flew the AA Flight 77 from
Dulles International Airport (Dulles) to Los Angeles International
Airport (LAX) on several occasions. had identified MOHAMED
ATTA as a person posing as a pilot seated in one of the jump seats
in the cockpit on one of his flights originating out of Boston. ATTA
had asked a lot of questions about the instrumentation and
specifically about changing course to New York. He also told
that he was going to mainline Delta to fly 767s. thought that
it was odd that a young pilot would go straight to flying 767s, since
those spots are typically reserved for those with seniority.
The first time had reviewed the document, he had not seen
any correlation between the equation on the document and piloting
a plane. However, after reviewing the theory by SA and SA
admitted that Diagram 2,
associated with the theory,
provided him more insight into the equation and that it could be used
to calculate a rate of descent when flying a plane.
Given a distance from a target, the altitude at that location,
and the current airspeed one could calculate the rate of descent and
plug it into the computer on the plane in order to initiate a descent
to that target. It would be reasonable to use the equation in the
document in order to calculate the descent rate of an aircraft.
All large airports have GPS plates located in their runways that
send a signal to an aircraft relaying the latitude and longitude (GPS
coordinates) of that runway. This signal provides pilots the
information they need to set an accurate descent rate to the runway
from a given location in the sky. However, if somebody wanted to
land a plane at a location where there are no GPS plates, they would
need to plug in a descent rate into the computer and they would need
to know either the GPS coordinates of that location or how far the
aircraft was from that location. With a known distance from a given
location and the altitude of the plane, a descent rate could be
calculated by the pilot. This calculation would have been done by
hand since there is not a way for the plane's computer to perform
this calculation. It is not likely that the use of the aircraft
computer for navigation would not have been known by a person who
had taken only minimal fight lesson, since that type of instruction
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EO14040-003666
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To: Counterterrorism From: San Diego
Re: , 04/05/2012
was only taught in advanced training. In commercial aviation, a
typical descent rate would be approximately 2,000 to 2,500 feet per
minute.
In 2001 it may have been difficult for an average citizen to
establish the latitude and longitude of a certain location. As a
pilot, had access to a Mark 6 plotting board- map, which provided
calculations based on a known location, where if the latitude and
longitude of a certain location was known, then one could calculate
the latitude and longitude of a nearby location using the this
board-map. was familiar with the flight path between LAX
and Dulles. Due to the air
traffic in the vicinity of Dulles, not
all planes taking off from Dulles would immediately climb to an
altitude of 35,000 feet. It was not unusual for commercial flights
out of Dulles to level off around 8,000 to 10,000 feet in that area
for as far as 100 to 150 miles, in order to avoid traffic. The
airspeed of a commercial airliner flying from Dulles to LAX would
be approximately 360 to 380 nautical miles per hour (knots) due to
head-winds experienced by an aircraft when flying west. However,
from LAX to Dulles,
a plane may experience tail- winds up to 120 to
140 knots, which would significantly increase the plane's airspeed
going from west to east (FBI note: was speaking of airspeed
and not speed over the ground, where airspeed is the speed of the
aircraft relative to the air, taking into consideration head- winds
and tail-winds which can be significant at high altitudes). At an
altitude of 8,000 to 10,000 feet, one would be able to see for
approximately 100 miles on
a clear day and even be able to identify
various landmarks.
As a pilot, had access to maps that had the mileage listed
on the legs between various waypoints along flight paths. These
waypoints typically had a latitude and longitude associated with
them. If one wanted to get the mileage (in nautical miles (nm)) along
a flight path, one could either add the mileage of the legs between
the waypoints or measure by hand the distance of the flight path.
If he were to calculate the distance along a flight path by hand using
these waypoints, he would have started from the beginning and not
worked backwards starting from the destination. When measuring a
distance by hand between waypoints on these maps, one could be
accurate within 0.5 nm or better.
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EO14040-003667
To: Counterterrorism From: San Diego
Re: , 04/05/2012
In a case where a pilot would have to change a plane's course,
such as on 9/11, the pilot would not necessarily need to disable the
autopilot for this type of course redirection, and could even utilize
autopilot after the change in course.
could not think of any other reason for the equation given
the parameters set forth on the document than to calculate a des cent
rate from a given altitude.
i http://mintaka.sdsu.edu.GF/explain/atmos_refr/horizon.html
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EO14040-003668