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requirements selected by the user to represent distances at which air traffic control
intervention would be either contemplated or provided, to determine air traffic control
separation assurance requirements.
The Intersection Model is not the final answer, but it is reasonable enough to provide a
partial answer, under a wide variety of cases. It is certainly preferable to a Monte Carlo
approach for the cases for which it applies.
An interesting thing to note is that while the Gas Law Model and the Intersection Model
are based on completely opposite assumptions about the structure of traffic, they have one
important factor in common. They both indicate that collision risk, all other things being
equal, is proportional to the square of the traffic level, n. This is common to most other
midair collision risk models. This comes from the fact that each of the n aircraft could, in
theory, collide with n-1 other aircraft, and thus the number of combinations is
C = n (n-1)/2
which is approximately proportional to the square of n. This suggests that while the
number of midair collisions has been very small in the past, the threat will grow out of
proportion to the increase in traffic. For example, a 50 percent increase in traffic would
correspond to a 125 percent increase in risk, all other things being equal.
5.3.3 Theoretical vs. Empirical Probability Distributions
Empirical Distributions
The above models produce an estimate of a mean value (risk) based on the expected value
of the independent variable (traffic density). If one airport had traffic activity that varied
much more than another with the same total traffic count, the estimated expected number
of collisions would be the same, but the actual risk could be greater. This can be
accommodated in a probabilistic model with a sample empirical probability distribution.
APPROACHES TO COLLISION RISK ANALYSIS
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For example, consider the Gas Law Model, with an assumed coefficient of C = 10 -12,
applied to a varying traffic load, expressed as an annual rate, following the distribution:
Frequency of Occurrence Operations per Year Expected Number of
Collisions
0.40 50,000 0.0025
0.30 100,000 0.0100
0.20 200,000 0.0400
0.10 350,000 0.1225
Expected value: 125,000 0.0242
In other words, the airport is operating at a rate of 50,000 operations per year 40 percent
of the time, etc. The average rate is 125,000 operations per year. The weighted expected
number of collisions per year is 0.0242. This is considerably larger, and presumably more
accurate, than the value of 0.0156 that one would obtain from the use of the annual
average alone.
A similar approach can be applied to the Intersection Model. Here the aircraft volume can
be divided into two streams and expressed in aircraft per hour. As a hypothetical sample,
consider an intersection where the internal crossing angle is 168 degrees. Path 1 is level
and path 2 descends at 3.2 degrees, passing 20 feet under path 1 at the intersection. The
aircraft (B737) on path 1 fly at 400 kts., have a wingspan of 193 ft., and a height of 20 ft.
The aircraft (PA31-350) on path 2 fly at 180 kts., have a wingspan of 44.5 ft., and a
height of 10 ft. Given that both paths have one aircraft per hour, the expected number of
blind flying collisions was computed to be N = 0.00041 per hour.
The collision risk is proportional to the product of the traffic volumes. Assume that the
volumes tend to be dependent with an empirical distribution given below:
Frequency of Occurrence Aircraft per Hour Blind Flying Collisions /hr
Path 1 Path 2
0.2 0.5 0.3 0.00006
0.2 0.5 0.9 0.00018
0.2 1.0 1.0 0.00041
0.3 1.0 1.2 0.00049
0.1 3.0 2.0 0.00243
Weighted Avg. 1.0 1.0 0.00052
The weighted average number of blind flying collisions would be 0.00052 per hour, rather
than the 0.00041 based on the average traffic density.
Theoretical Distributions
The above are examples of the use of discrete, empirical probability distributions. A
number of models have been constructed using theoretical distributions. Some of the most
SEPARATION SAFETY MODELING
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popular distributions are the Normal (or Gaussian), the negative exponential, and the
Poisson. The first is continuous, symmetrical, and has infinite tails. The second is
continuous, produces non-negative values, and has an infinite tail. The third is a discrete
distribution with positive probability values at non-negative integers and with an infinite
tail.
One of the advantages of theoretical distributions is that they can be manipulated
analytically. Another is that an assumed distribution and a few parameters estimated from
empirical data can produce probabilities for an infinite number of values, whereas an
　
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