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there clearly are deficiencies in the methodology. First, aircraft are not distributed
randomly in a terminal area. Even if they were so distributed, the model would only make
sense if X represented the number of aircraft in the given airspace at the same time. (An
aircraft in the airspace on March 1 will not come into conflict with one in the airspace on
July 17!) Apparently the value used for C was chosen to “adjust” for using annual
operations counts, but as seen in the following example, the “adjustment” was deficient.
The model suggests that the collision risk at Chicago, O'Hare International (ORD) with
about 900,000 operations per year would be 300 times as great as Santa Paula (California)
Airport (SZP) with about 52,000. But Santa Paula had four midair collisions between
1983 and 1997 (the most for any US airport) while O'Hare had none. Perhaps the
operating conditions were not the same at both airports, the aircraft were not really flying
randomly (particularly at ORD), and/or the use of annual traffic counts was improper. (A
second example is presented in Section 5.3.3.)
The Intersection Model
A complete opposite of the Gas Law Model is the intersection model, which assumes that
all aircraft fly on straight-line paths, at constant speed, and in level flight. Because not all
aircraft fly from the same origin to the same destination, these paths often cross at (not
necessarily published) intersection points. On the face of it, this is a much more realistic
model in that aircraft tend to fly fixed routes at constant altitudes for much of the time.
APPROACHES TO COLLISION RISK ANALYSIS
5-11
Over very short distances, these paths can be reasonably approximated by constant speed
and direction.
If the placement of an aircraft on path 1 can be assumed to be uniformly random (which is
reasonable if there is no reason why the aircraft should more likely be at any given point
on the path at a given time than any other point), then the probability of a given aircraft on
path 2 colliding with the aircraft on path 1 is given by
P = 2m1 R(v1
2 - 2v1v 2 cos ø +v2
2) ½ / v1 v2 sin ø ,
where
P = probability of a given aircraft on path 2 colliding with any one of the
aircraft on path 1,
m1,2 = number of aircraft on path 1 or 2 per hour,
v1,2 = velocity of aircraft of aircraft on path 1 or 2 in knots,
R = average size of aircraft in nm.,
ø = (interior) crossing angle in degrees ( 0 < ø < 180).
The model assumes that the aircraft are circles with R equal to the sum of the radii.
If the placement of the aircraft on the paths can be assumed to be independent (which is
reasonable in the en route airspace for aircraft going between different origins and
destinations), and the traffic densities remain constant, then the expected number of
collisions at the intersection can be approximated by
N = m2 P .
This result was presented by W. Siddiqee in “A Mathematical Model for Predicting the
Number of Potential Conflict Situations at Intersecting Air Routes” [R5.17]. This model
was developed by assuming that the aircraft are uniformly distributed on their respective
paths and that all aircraft on the same path fly the same speed. A similar result might be
obtained under other assumptions, so long as the average speeds and the average number
of aircraft per hour are used. The model also assumes, as does the Gas Law Model, that
there is no intervention, that is, aircraft on collision courses remain on collision courses
(“blind flying”).
Use of this model does not require the assumption that all aircraft in the airspace are on
either of two paths. It can be used repeatedly on a large number of intersections, and for
various traffic loadings and speeds.
This model was generalized by K. Geisinger in “Airspace Conflict Equations” [R5.19] to a
three dimensional model, allowing either or both paths to be climbing or descending. The
aircraft are modeled as discs, with R the sum of the radii and Y the average height of the
two aircraft. The paths are also allowed to cross each other with a vertical path
SEPARATION SAFETY MODELING
5-12
separation, H, at the point of intersection in the horizontal plane. When one or both paths
are not level, it is possible to have a collision even when H>Y, because the collision need
not occur at that point. The equations are too complex to reproduce here.
These equations were incorporated into the Sector Design Analysis Tool (SDAT), a
computer tool for evaluating airspace design. It automatically determines where paths
cross, based on actual recorded radar data. The computations are based on separation
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a concept paper for separation safety modeling(25)
