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empirical distribution is limited to the discrete values that have been observed. This is also
a disadvantage, especially in the all important issues of the tails. It is typical that collision
risk is heavily influenced by the values that are far out on the tails.
Even if the empirical data fit a theoretical distribution very well near the mode, it is by no
means certain that the distribution’s tails are an appropriate representation. For example,
assume that flight deviation from an assigned altitude of 35,000 feet seems to fit a Normal
distribution very well with a set of observations that show deviations up to 500 feet above
and below the assigned altitude. Use of the Normal distribution would result in a certain
probability that the aircraft could be at 60,000 feet or more above or below 35,000 feet.
The usual way out of this is to truncate the distribution, with the truncation occurring at
some points above and below which the values are deemed unrealistic. The Reich Model
[R5.1, R5.2], for example, uses truncation points. But how should these truncation points
be determined?
Another problem is that variations in the observed data might be due to a number of
different, unrelated causes. For example, altitude deviations of a few hundred feet might
be due to sudden changes in wind speed or barometric pressure; deviations of a few
thousand feet might be due to miscommunication of assigned altitude; and greater
deviations might be due to serious pilot or aircraft problems. Each has its own probability
of occurrence and characteristics. The common practice of assuming a single theoretical
distribution to cover all might (or might not) be an over simplification. A similar problem
arises in modeling horizontal deviations from nominal routes. The same problem can
occur in discrete distributions.
One way of dealing with this situation is to derive a “mixture distribution” which is
obtained as a weighted sum of the probability density functions of two or more
distributions. A theoretically-based model with horizontal track deviations based on the
weighted sum of two different Normal distributions (one for navigational errors and one
for blunders) is presented in [R5.18; pg. 33].
A bounded, discrete distribution can be based either on values from a truncated theoretical
distribution or from empirical measurements. An empirical distribution has finite tails,
which are arbitrary, for a few more samples might reveal an even more extreme point.
One advantage of an empirical distribution is that it reflects the observed, real world,
including gaps, lack of symmetry, multi-modality, etc., that may implicitly represent many
APPROACHES TO COLLISION RISK ANALYSIS
5-15
unknown factors. Some apparent quirks might have causes that should be considered
explicitly.
In closing, we note that Monte Carlo simulation models, may employ sampling from
empirical or theoretical distributions.
5.3.4 Intervention Considerations
The Intersection Model described above produces an estimate of collision risk at the single
intersection point of two flight paths. An airspace region contains numerous path
intersections, which modeling tools, such as SDAT, can use to estimate collision risk,
controller workload, and other metrics. For example, a study conducted using SDAT on
sample data from three different ARTCCs, suggested that the blind-flying collision risk
amounted to approximately one midair collision per month in the airspace controlled by
each ARTCC.
Fortunately, aircraft do not “fly blind.” Flight crews are often able (through visual
sighting or TCAS alerts) to detect a pending conflict and take appropriate corrective
action. In the positive control environment, air traffic control provides separation for all
air traffic. The first line of defense is the ATC system, backed up by the flight crews. It is
always possible, although extremely rare, that a conflict will proceed undetected until it is
too late to perform a maneuver to prevent it. Operational error and pilot deviation reports
document cases of this.
In order for a collision to occur, the conditions that would lead to a blind flying collision
are required along with a situation in which neither the ATC system nor the flight crews
detect and correct the threat in time to prevent it. If the conditions that would lead to a
blind flying collision are present, it takes a series of events, each requiring an uncertain,
i.e., stochastic, amount of time, to avoid the collision:
Either:
1. The controller detects the threat first;
and
a. The controller orders the first aircraft to maneuver, and
b. The pilot of the first aircraft interprets the controller’s communication and moves the
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a concept paper for separation safety modeling(27)
