NBSIFS 30-1990

-At 3r3r3r&4--2J R U U t; Probabilities of Vertical

A Sensitivity Analysis

Overlap:

$

Howard K. Hung, Judith F. Gilsinn, and Karla L. Hoffman

Operations Research Division

Center for Applied Mathematics

National Bureau of Standards

U.S. Department of Commerce

Washington, D.C. 20234

June 1979

Issued March 1 980

Working Paper for

Federal Aviation Administration

Office of Systems Engineering Management

800 Independence Ave., S.W.

Washington, D.C. 20591

100

,U56

80-1990

1380

NBSIR 80-1990

PROBABILITIES OF VERTICAL OVERLAP:

A SENSITIVITY ANALYSIS

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OF STANDARDS

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Howard K. Hung, Judith F. Gilsinn, and Karla L. Hoffman

Operations Research Division

Center for Applied Mathematics

National Bureau of Standards

U.S. Department of Commerce

Washington, D.C. 20234

June 1979

Issued March 1980

Working Paper for Federal Aviation Administration

Office of Systems Engineering Management

800 Independence Ave., S.W.

Washington, D.C. 20591

U.S. DEPARTMENT OF COMMERCE. Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Science and Technology

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director
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TABLE OF CONTENTS

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1. Introduction 1

2. Probability of Vertical Overlap 3

3. Factors Affecting Probability of Vertical Overlap 6

3.1 Probability Distributions 6

A. Normal Distribution 9

B. t Distribution 10

C. Normal/Double-Exponential Distribution .15

3.2 Standard Deviations 19

3.3 Aircraft Vertical Dimensions 19

3.4 Vertical Separation Standards 20

4. Description of Computer Programs 23

5. Results of Computation 25

5.1 Normal Distribution 25

5.2 t Distribution 29

5.3 Normal/Double-Exponential Distribution 41

6. Discussion and Conclusions 58

7. References 62
LIST OF TABLES

Table 1 Comparison of analytical results with computer runs under normal distributions 30

Table 2 Probabilities of vertical overlap under t distributions

for three different separation standards 39

Table 3 Values of and a^

DE

for certain values of y, p, and a^..42

Table 4 Values of input parameters for figures 18-29 43

Table 5 Probabilities of vertical overlap under normal /doubl e- exponential distributions for three different separation

standards 44
-n iLIST OF FIGURES

Figure 1 Probability density functions of normal, Cauchy, and

normal /double-exponential distributions 8

Figure 2 Probability density functions of the normal distribution for three standard deviations H

Figure 3 Probability density functions of the normal distribution for three standard deviations, on vertical

logarithmic scale ...12

Figure 4 Area of vertical overlap between two aircraft for

normal distributions and two standard deviations 13

Figure 5 Probability density functions of the t distribution

for three standard deviations 16

Figure 6 Probability density functions of the t distribution

for three standard deviations, on vertical logarithmic

scale : 17

Figure 7 Probability density functions of the normal /doubleexponential distribution for three standard deviations ...21

Figure 8 Probability density functions of the normal /doubleexponential distribution for three standard deviations,

on vertical logarithmic scale 22

Figure 9 Probability of vertical overlap vs. separation standard

for a normal distribution, aircraft vertical dimension

of 30 feet, and three standard deviations 26

Figure 10 Probability of vertical overlap vs. separation standard

on a vertical logarithmic scale, for a normal distribution, aircraft vertical dimension of 30 feet, and three

standard deviations 27

Figure 11 Probability of vertical overlap vs. separation standard,

for a normal distribution and aircraft vertical dimensions

of 30 and 50 feet 28

Figure 12 Probability of vertical overlap vs. separation standard,

for t distribution with 20 degrees of freedom, aircraft

vertical dimension of 50 feet, and three standard deviations 33
-IVFigure 13

Figure 14

Figure 15

Figure 16

Figure 17

Figure 18

Figure 19

Figure 20

Figure 21

Figure 22

Probability of vertical overlap vs. separation standard,

for t distributions with 40 degrees of freedom, aircraft

vertical dimension of 50 feet, and three standard deviations 34

Probability of vertical overlap vs. separation standard,

for t distributions with 3, 6, and 9 degrees of freedom,

aircraft vertical dimension of 30 feet, and standard

deviation of 100 feet 35

Probability of vertical overlap vs. separation standard,

for t distributions with 20, 30, and 40 degrees of freedom,

aircraft vertical dimension of 30 feet, and the standard

deviation of 100 feet 36

Probability cf vertical overlap vs. separation standard,

for a t distribution with nine degrees of freedom, and

aircraft vertical dimensions of 30 and 50 feet 37

Probability of vertical overlap vs. degrees of freedom,

for t distributions, standard deviation of 100 feet,

aircraft vertical dimension of 30 feet, and three separation standards 38

Probability of vertical overlap vs. separation standards,

for a normal/double-exponential distribution with p = 0.95

and y

= 1, aircraft vertical dimension of 30 feet, and

three standard deviations 46

Probability of vertical overlap vs. separation standard,

for a normal/double-exponential distribution with y

= 0.95

and p = 4, aircraft vertical dimension of 30 feet, and

three standard deviations 47

Probability of vertical overlap vs. separation standard,

for a normal/double-exponential distribution with y

= 0.95

and p = 7, aircraft vertical dimension of 30 feet, and

three standard deviations 48

Probability of vertical overlap vs. separation standard,

a normal/double-exponential distribution with y

= 0.95 and

p = 1, aircraft vertical dimension of 50 feet, and three

standard deviations 49

Probability of vertical overlap vs. separation- standard,

for a normal/double-exponential distribution with y

= 0.95

and p = 4, aircraft vertical dimension of 50 feet, and

three standard deviations 50
-vFigure 23 • Probability of vertical overlap vs. separation standard,

for a normal /double-exponential distribution with y

= 0.95

and p = 7, aircraft vertical dimension of 50 feet, and

three standard deviations ..51

Figure 24 Probability of vertical overlap vs. separation standard,

for a normal /double-exponential distribution with y = 0.99

and p = 1, aircraft vertical dimension of 30 feet, and

three standard deviations ...52

Figure 25 Probability of vertical overlap vs. separation standard,

for a normal/double-exponential distribution with y = 0.99

and p = 4, aircraft vertical dimension of 30 feet, and

three standard deviations ...53

Figure 26 Probability of vertical overlap vs. separation standard,

for a normal /double-exponential distribution with y = 0.99

and p = 7, aircraft vertical dimension of 30 feet, and

three standard deviations ...54

Figure 27 Probability of vertical overlap vs. separation standard,

for a normal/double-exponential distribution with y

= 0.99

and p = 1, aircraft vertical dimension of 50 feet, and

three standard deviations ...55

Figure 28 Probability of vertical overlap vs. separation standard,

for a normal /double-exponential distribution with y

= 0.99

and p = 4, aircraft vertical dimension of 50 feet, and

three standard deviations ...56

Figure 29 Probability of vertical overlap vs. separation standard,

for a normal/double-exponential distribution with y

= 0.99

and p = 7, aircraft vertical dimension of 50 feet, and

three standard deviations ...57
-ViAbstract

Because of the potential increase in traffic at FL 290 and above, both

current and alternative vertical separation standards are being reviewed.

A plan to collect data on the vertical navigational performance of aircraft

is also contemplated. This report documents a sensitivity analysis carried

out to assess how different assumptions about the probability distribution

of “total vertical error" affect estimates of the probability of vertical

overlap. The four factors affecting the probability of vertical overlap

which are examined in this study are: shape of the vertical -error distribution; the standard deviation (and other parameters, if any) of this

probability distribution; the vertical dimensions of the aircraft; and the

vertical separation standard. Probabilities of vertical overlap were

computed over a range of possibilities for each of these four factors in

order to discern the effect of each factor. A final section discusses

the findings of this study and draws some conclusions.

Key Words : Air safety; collision probability; probability distribution; " — sensitivity analysis; separation standard; vertical error;

vertical overlap.
- 1 -

1. Introduction

In its continuing effort to improve the National Airspace System, the

Federal Aviation Administration (FAA) is evaluating the existing standards

for separation of designated parallel flight tracks in the airspace over

conterminous United States (CONUS). As part of this effort, the FAA intends

to study the implications of both current and alternative vertical separation

standards for aircraft operating over CONUS at flight levels of 29,000 feet

(FL290) and above. The paucity of existing data on the vertical component

of navigational performance necessitates initiating a data collection program

to determine the altitude- keeping capabilities of aircraft flying at FL290

and above. The FAA is in the process of designing a plan for the collection

and analysis of these data.

To support the FAA in this effort, the Operations Research Division of the

National Bureau of Standards has been investigating analytical problems

relating to aircraft vertical separation standards [l].

1

This report presents

the results of one such study, a sensitivity analysis to assess how different

assumed probability distributions of "total vertical error" affect the probability of vertical overlap.

? The total vertical error is defined as the difference between the assigned

3

pressure altitude of an aircraft and its true pressure altitude. A vertical

^Numbers in the square brackets refer to the References listed at the

end of this report.

2

Total vertical error, as defined in Reference [1], is the sum of the pilot

response error, the altimeter instrument error, and the static pressure

system error.

3

The altitude measured by an altimeter is termed pressure altitude here

because it is first measured barometrical ly in terms of atmospheric pressure,

and then converted to altitude.
- 2-

overlap occurs when the vertical separation between the centroids of the

rectangular envelope of a pair of aircraft is less than the average of the

vertical dimensions of the aircraft. It should be emphasized that an overlap

is not a collision, and that the probability of vertical overlap is not the

probability of collision. A collision occurs only when the pair of aircraft

overlap in all three dimensions at the same time. This report will examine

the likelihood of overlap in only one dimension--the vertical dimension.

The four factors affecting the probability of vertical overlap which are examined in this study are: the distributional form of the probability density

function for total vertical error, the standard deviation of this probability

density function, the vertical dimensions of the aircraft, and the vertical

separation standard. Probabilities of vertical overlap were computed over a

range of possibilities for each of these four factors in order to discern

the effect of each factor. The computational results have been plotted to

permit visual inspection.

The remainder of this report is organized as follows. Section 2 reviews the

definition of probability of vertical overlap between two aircraft, and Section

3 discusses the four factors affecting this probability which were studied.

Section 4 briefly outlines the approach employed to compute the probability

of vertical overlap. Section 5 reports the computational results. Finally,

Section 6 discusses various findings of this study and draws some conclusions.

The report did not use metric units because in the fields of air traffic control

and airspace management currently conducted in the U. S., the aircraft altitude

is expressed either in feet or in hundreds of feet. A conversion of the altitude measurement system to metric units is not likely in the near future [9].
-3-

2. Probability of Vertical Overlap

The approach employed here to define the probability of vertical

overlap is based on the collision risk methodology pioneered by P. Reich

[2], which assumes that all aircraft in the track system are of the same

size,^ and which represents each aircraft as a rectangular slab with

longitudinal, lateral, and vertical dimensions of x , x , and x corresponding to the length, width, and height of the aircraft, respectively.

Two aircraft are said to overlap in the rth dimension (where r can be

x, y, or z) when the separation between the centroids of the two aircraft

in the rth dimension becomes less than X r

- A collision is considered to

occur when a pair of aircraft overlap in three dimensions simultaneously;

in other words, when the distances between the centroids of the two aircraft

longitudinally, laterally, and vertically are less than x x

, X , and x z#

respectively.

Since the probability of vertical overlap is the main subject of this

report, we will explore this concept in detail.

Assume two aircraft are flying over the same geographic point on the

earth. The vertical overlap probability between them is the probability

that the centroid of the second aircraft ("2 ,! ) is separated vertically by

less than one aircraft height (the vertical dimension) from that of the

first aircraft ("1"). Formally, we have

4

If the sizes of a pair of aircraft differ in some dimension, the average

of their sizes in that dimension can be used in the pertinent mathematics

without loss of validity.
-4-

is centered at v and "2" is in (v-A

z

, v+A

z

)}

where A z is the aircraft vertical dimension, v ranges over the possible

vertical positions of the first aircraft, and P{ } is the probability of

the event in brackets.

If (as is assumed here) the two aircraft are flying independently, the above

equation can be restated as follows:

Assume f^(v) and f^(w) are the probability density function of total

vertical error for aircraft 1 and aircraft 2, respectively. The probability

that the centroid of aircraft 1 lies between altitudes v and v+dv, relative

to its assigned altitudes, is f,(v)dv. Since the two aircraft are nominally

separated by a separation standard S in the vertical dimension, the probability that the altitude u of aircraft "2" (relative to aircraft 1 '

assigned altitude) lies in the interval between v-A

z and v+A 7 is the same

as the probability that its vertical error w=u-s lies between (v-A )-s and

(v+A

z )-s

PCI" is at v} P{"2 !

’ is in (v-A , v+A

z )}

V+A

Z

z

The probability that "1" is at v and the two aircraft overlap is
-5-

f

1 (v)dv

V-A

z

f^lu-Sjdu.

By integrating over all values of v, the probability of vertical overlap

can be represented as follows

P{Vertical Overlap} = P

Z (S) = /

'!<»

—00

V+A

z

z

f2(u-S)du dv;

cf. References [1] - [2].

Throughout this paper we will assume that the probability distributions

of vertical flight error are the same for both aircraft, i.e., f^ = f^.

Although it is possible to relax this assumption, the analysis becomes

considerably more complicated, and there is no reason to believe that

different distributions are likely.
- 6

-

3. Factors Affecting Probability of Vertical Overlap

From the above discussion, it is clear that the probability of vertical

overlap of a pair of aircraft depends on the following factors:

(1) The shape of the probability density function (f^ = f

2 = f)

for total vertical error of an aircraft.

(2) The parameter(s) that specify the probability density

function (e.g. the standard deviation).

(3) The vertical dimension (x ) of the aircraft.

(4) The vertical separation standard (S).

We shall discuss each of these factors in the following pages, and present

computed results on their effects upon overlap probabilities.

3.1 Probability Distributions

Because of the lack of empirical data, the mathematical form of the

probability distribution for total vertical errors is not well established.

Although it is generally accepted that this probability distribution is

unimodal and symmetric [3], debate over its shape continues. The

distributional form is expected to affect significantly the probability of

overlap. Employing the assumption that the probability of total vertical error

is unimodal and symmetric, we limit our investigation to such probability

distributions, differing only in "tail shape." It is hypothesized by many

in the air traffic safety field and supported by some empirical data on

navigational performance that small vertical flight errors (the majority of

the errors) are normally distributed, while the larger errors (rare events)

follow a double exponential distribution [4], The resultant probability

distribution is a weighted combi naci on of a normal distribution and a double exponential distribution.

*A unimodal function has exactly one peak or "bump."
- 7

-

Another probability distribution family which we will include in our

analysis is the t distribution. A t distribution is totally described

by one parameter, the number of degrees of freedom. A t-distribution

with one degree of freedom is a Cauchy distribution, while a t-distribution with infinite degrees of freedom is a normal distribution. Varying

the degrees of freedom of this distribution while keeping the standard

deviation fixed, alters the shape of the density function.

Therefore, we will investigate three different probability distribution

families: the normal distribution, the t distribution, and the weighted

normal/double-exponential distribution. These include a wide range of unimodal

and symmetric probability forms. Regarding them as possible probability

distributions of total vertical errors, we will examine how these functional

forms and corresponding parameter values affect the probability of vertical

overlap. The probability density functions of the normal distribution,

the t distribution, and the normal /double-exponential distribution are

plotted in Figure 1.
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(A) Normal Distribution

The normal distribution is the most widely known and used of all

distributions. It arises as the distribution for measurement or

control errors in many physical processes.

The normal distribution, symmetric about 0 (i.e., positive and negative

errors are equally likely), has a bell-shaped probability density function

given by

f(x) exp - 00 < x < °°

,

where a is the standard deviation of the distribution. Note that log f(x)

is a quadratic function of (x/a), explaining the parabolic shape of the

solid curve in Figure 1.

In the air traffic environment, large flight errors are of great concern

despite their rarity, since these rare errors may lead to midair collision or near-collision. For a normal distribution, the probabilities

that the random variable deviates from its mean by less than one standard

deviation, two standard deviations, and three standard deviations are

respectively 0.6826, 0.9544, and 0.9974. The probability of deviating from

the mean by more than three standard deviations is only 0.0026.

Normal probability density functions with standard deviations of 100

feet, 150 feet, and 200 feet are plotted in Figure 2 and Figure 3. To help

display the characteristics of the tail portion of normal probability distributions, a logarithmic scale is used for the vertical axis in Figure 3.
- 10 -

Figure 4 illustrates the vertical overlap between two aircraft, each

having a normally distributed total vertical error, when the vertical

separation is set at 1000 feet. Their assigned altitudes are 30,000

feet and 31,000 feet, respecti vely . The dotted lines represent normal

distributions with a standard deviation of 150 feet and the solid lines

represent normal distributions with a standard deviation of 200 feet.

The overlaps occur in the shaded area centered at 30500 feet on the

horizontal axis. The shaded area under the solid line is the area of

overlap when the standard deviation is 200 feet. One can see in Figure

4 that the probability of overlap when the standard deviation is 150

(the dotted line) is much smaller. All these overlaps occur in the tails

of normal probability distributions.

(B) t Distribution

The density function of the simplest t distribution with n degrees of

freedom, is given by

f(t)

rT(n+l)/2~]

n-rr r(n/2)

oo < t < c°

where r(*) is the gamma function. For non-negative integers m,

r(m+l) = m! and r(^r-) = . .

. \ . /tt . This probability

density function is symmetric about its mean of zero, and its standard

deviation is (pA?)h for n>2. It is customary to describe the

characteri sties of this distribution in terms of the number of degrees

of freedom. If t is replaced by ^-p— , where y and k are
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FT,

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-14-

constants, then the resulting transformed t distribution would have

*

ct mean of p and standard deyi^tion of (- n /n~2. )„. i n the context of

aircraft vertical navigational performance, the p is the assigned altitude

and k is a conversion factor to show the size of the standard deviation

of the t distribution. We consider the t distribution because for small

degrees of freedom, Tt has larger tails than does the normal distribution.

When the number of degrees of freedom is one, the t distribution is the

Cauchy distribution which has extremely long tails; as the number of

degrees of freedom approaches infinity, the t distribution approaches a

normal distribution. By varying the degrees of freedom of the t distribution while keeping the standard deviation fixed, we obtain many distributions with tail characteristics between those of a normal distribution

and those of a Cauchy distribution.

Probability density functions of the t distribution with nine degrees of

freedom and standard deviations of 100 feet, 150 feet, and 200 feet are

plotted in Figures 5 and 6. Figure 5 depicts the shapes of the graphs

of these functions while Figure 6 magnifies their tail portions. It is

interesting to note from Figure 3 that on log paper normal probability

density functions are concave curves, whereas Figure 6 shows that the

corresponding "log" graph of a t probability density function can be

dissected into two convex curves if one ignores that part of the graph

near the mean, between the inflection points at t = + n

2

. This suggests

that as one moves farther away from the mean, the t probability density

function decreases more slowly than that of the normal distribution with

the same standard deviation, indicating a longer tail. This relationship

suggested by Figures 3 and 6 can be proven analytically; we omit the details.
- 15 -

(c) Combination of Normal and Double Exponential Distributions

It is hypothesized by many in the air traffic safety field, and supported

by some empirical data on navigational performance, that small flight

errors (the majority of the errors) are normally distributed while the

larger errors (i.e., the rare events) follow a double exponential

distribution. The resultant probability density function is a weighted

combination of those for a normal distribution and for a double exponential distribution. Since the density function of a normal distribution

is

and the density function of a double exponential distribution is

where a = mean andJ2s = standard deviation, the density function of a

combination of normal and double exponential distributions is

00 < x < 00

00 < x < 00

g(x) = Y f

x (x) + (1-y) f

2

(x) 00 < X < 00

where 0 <_y <_ l.
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-18-

Let and cr^ be the standard deviations of the normal and the double

exponential distributions respectively, i.e., the a and /2b- of the preceding

equations. Then the standard deviation of a combination of normal

and double exponential distributions is

°NDE

= (ya

N

2 + <W>»DE)

' i

’ 0 - Y - 1 •

Since it is likely that the sources of vertical flight errors reflected in

the normal distribution are different from the sources of the rare events,

it is reasonable to assume that the standard deviations for normal and

double exponential distributions are generally of different values,

p = a^fr/a^ will be used to denote the ratio of these two standard deviations.

Hence, the relationship between the standard deviation of the normal/doubleexponential distribution and that of the normal distribution can be represented

as follows

NDE l Y0

N

2 + (i-y >°de2 *

Ya M

2 + ( 1 -^)p 2

ct M

2

Y + (1-y)p 2

The probability density functions of a combination of normal and double

exponential distributions with y

= 0.95 and p = 7.0, and with respective

values of 100, 150, and 200 feet for the standard deviation of the normal

distribution, are plotted in Figure 7 and Figure 8.
- 19-

The probability distributions of normal /double-exponential type plotted

in Figures 7 and 8 are combinations of 95% normal distribution and 5%

double exponential distribution. The body of the combined distribution

is close to that of a normal distribution, while its tail is close to that

of a double exponential distribution. This phenomenon is explicitly

shown in Figure 8 where a logarithmic scale is used on the vertical axis.

3.2 Standard Deviations

It is generally true that the standard deviation of a probability distribution largely determines its shape "within" the over-all functional form

involved. The shape of the probability distribution of the total vertical

errors is important in estimating the probability of vertical overlap,

because most overlaps are supposed to occur in the tail of the probability

distribution. Also, the standard deviation has a significant interpretation

as a measure of the precision of altitude control. Therefore, it is

important to examine the potential impact of the standard deviation on the

probability of vertical overlap. Standard deviations of three values;

namely, 100 feet, 150 feet, and 200 feet are used in most computer runs in

this study. These values are within the range of values of standard deviations of vertical errors found in an early study [5].

3.3 Aircraft Vertical Dimensions (Heights)

A vertical overlap of a pair of aircraft occurs when the centroid of the

second craft is within one aircraft height of the centroid of the first.

Thus, the height of the two aircraft will affect the vertical overlap

probability. There are many different types of aircraft flying above

29,000 feet over CONUS. In this study, we examined two representative
- 20 -

aircraft heights: one was = 30 ft. which is roughly that of a mediumsized jet airliner, and the other was x z = 50 ft., the approximate height

of a large-sized jet airliner [6].

3.4 Vertical Separation Standards

One of the most crucial factors influencing the vertical overlap probability

is the vertical separation standard. As discussed in Section 2, the probability of vertical overlap can be expressed as follows:

f

2

(u-S) du dv

where S is an assigned separation standard. The vertical overlap probability can be made infinitesimal by assigning an extremely large separation standard in the air traffic track -system. However, in so doing, the

traffic capacity of the track system might be reduced to an intolerable

level. The present vertical separation standard for flight levels above

29,000 feet over the CONUS is 2000 feet. In this study, we shall calculate

vertical overlap probabilities between a pair of aircraft for separation

standards ranging from zero to two thousand feet.

PIVertical Overlap} = P (S) = r° /*v+x - L ""M.
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SIGMA=

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- 22-

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-23-

4. Description of Computer Programs

Three computer programs, one for each of the three probability distributions discussed earlier, were developed. It was assumed that the

probability distributions of total vertical errors were the same for

the two aircraft involved. The vertical locations for the first and

the second aircraft, relative to the assigned height of the first, are

assumed to be at v and u=S+w feet respectively where v and w are random

variables and S is a given separation standard. Since standardized

distributions are used in the subroutines of the computer programs, the

necessary scaling of standard deviations is carried out in the main program.

The computer program first finds the probability that the centroid of

the first aircraft is at a particular altitude relative to its assigned

vertical location, and then calculates the probability that the centroid

of the second aircraft is within one vertical aircraft dimension of that

of the first. The product of these probabilities is the vertical overlap

probability of these two aircraft at that vertical location.

The formula for the probability of vertical overlap was derived earlier as

PIVertical Overlap} = P

z (S) -C f-j(v)

v+x

•/ —oo
- 24 -

This can be rewritten as

P z (s) =

OO ^V+X z”^ f

2

(v-x

z -s) dv

where F£(*) is the cumulative distribution for the probability density

function f2 (*)

Numerical integration was performed using the trapezoid rule to obtain

the probability of vertical overlap. Instead of integrating from minus

infinity to plus infinity, the computer program uses only the locations

between minus and plus six standard deviations from the center, and the

r i i n mesh size for the trapezoid rule is

qqq

The summation of overlap

probabilities at these locations is a good approximation to the total

probability of vertical overlap of the two aircraft.
-25-

5. Results of Computation

5.1 Normal Distribution

Probabilities of vertical overlap were calculated under the assumption

that the total vertical errors were normally distributed for each aircraft.

Two vertical dimensions of aircraft, 30 feet and 50 feet, and three sets

of standard deviations, 100 feet, 150 feet, and 200 feet, are used in the

computation. The separation standards examined in this study ranged from

0 to 2000 feet with an increment of 50 feet. The calculated results are

plotted in Figures 9 through 11. Figures 9 and 10 refer to the same set

of input data but use different scales on their vertical axes in order to

highlight respectively the vertical overlap probabilities below and above

the separation standard of 500 feet. Figure 11 shows the plots of probability of vertical overlap for two values of the vertical dimension of the

aircraft.

As observed in Figure 9 and 10, with increasing separation standards, the

probabilities of vertical overlap drop more rapidly under a normal distribution with a smaller standard deviation than under one with a larger standard

deviation. It is also observed that the curves of probabilities of vertical

overlap cross each other somewhere between the separation standards of 200

feet and 300 feet. This phenomenon will be discussed later in Section 6.

Since the convolution of two normal distributions is a normal distribution,

the probability of overlap of the two aircraft can be determined analytically

by integrating the convolved normal distribution. Since a normal probability
FAA

UERTICAL

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PROJECT

- 26 -

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FAA

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- 27 -

clq:o«<icqh >jhe-> OLl. ODLiJQiJCQ.

SEPARATION

STANDARDS

(FT)

NORMAL

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CLC£oec><xccit-i_ji-it-:>- OL O Z> Ld CE —1 <r ll

SEPARATION

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NORMAL

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30,

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!H=

50

FT
-29-

density function cannot be integrated in closed form, tables of Normal

Probability Functions [7] were used to obtain the numerical results

of the integration. Computational results obtained by the numerical

method and by the computer program are tabulated in Table 1. Column C/A in

Table 1 shows the relative magnitudes of the probability of vertical overlap

obtained by computer program (C) and by numerical methods (A). It can be

seen that the computed values are generally less than the numerically derived

-5

values for probabilities equal to or less than 10 . It is also observed

that as the values for probabilities decrease, the values for C/A also deviate

away from 1. Two possible explanations for the causes of the above phenomena

are (1) the computer program is limited to integrate to +6 sigma and (2) for

very small values, computer round-off errors can be significant. It should

be mentioned here that the Tables of Normal Probability Functions carry

only fifteen significant digits after the decimal point. This also

contributes to the discrepancy between the values obtained from the computer

program and that from the Tables of Normal Probability Functions.

Table 1 also serves as a summary of computational results of probabilities

of vertical overlap based upon normal distributions with three different

standard deviations, two aircraft vertical dimensions, and three separation

standards.

5.2 t Distribution

Probabilities of vertical overlap derived from the t distribution are

affected by four factors; namely, the number of degrees of freedom, the

standard deviation, the vertical dimension of the aircraft, and the

separation standard. Sensitivity studies have been carried out to assess

the affect of each of these four factors on the probability of vertical

overlap; the computational results are plotted in Figures 12 through 17,

which are discussed below.
< 1 1 ,“H ’ co

<_> 1 1 r^» 1 1 r^.

CXI r-H

r-H t-H ij r-H O O CO o O O 00

CX_J UO t-H o LO i cn o cn o • oCXI

II o r-H t-h CO •r— r-H +j,

1 >ec ao o O o on O O r-H CO O c CXI un < r-H CM O CO un o <=c . CO CTi , co o rv. cn P". o o 1 • • 1 T—

r-H t-H

r-H t-H VO

cn CO 1 1 vo

!=<_> o ^ un

O CO CO Ov_ CTi CXI o un

C_> co cn r-H CM o r-H CO r-H • • un

r-H 1 f "

ii

u t-H pH

oo •r— t-h r^. pH VO — 1 1 i i >«C i O CO CM CO O CO CO O CM O'! fO co un un un c co cn cn cm C r-H <*Q CO r-H • • • •

un cn o r-~ cn o < cn o r-H cn o i''. cn o oo cn o o • • • • • • r-H t-H

r-H t-H r-H LO CO r-H un CO

VO r-H r-H '—1 'd* CO

fc=c_> CM

N Cl

O oo 0.3. CO O CO r-H CO 00 o LO CO CO un co co o CM t-H r-H r— co cm o • • •

r-H

it

u t-H t-h 00 •r— <-h Ln co r-H LO CO i i i 1 1 1 >< CO r-H r^. O CO CM

r-sw* CO CO CO o <T3 O O co co cn oo C CO 00 CO r-H OO CO C CO r-H r-H cn co cm O O O | o o o D o un o o un o r-H r-H CXI '—1 r—1 CM o I o CO un

II ii N | N

r . r<

Table

1:

The

Probability

of

Vertical

Overlap

Obtained

by

Analytical

Method

and

Computer

Runs

Under

Normal

Distributions
-31-

Holding the number of degrees of freedom at 20 and 40, and the vertical

dimension of aircraft at 50 feet, probabilities of vertical overlap are

plotted for each of the three values (100, 150, and 200 feet) of the standard

deviation. Probabilities of vertical overlap derived from the t distribution with 20 degrees of freedom are plotted in Figure 12; those for 40

degrees of freedom are plotted in Figure 13. The difference between a curve

in Figure 12 and the corresponding curve in Figure 13 demonstrates the effect

of the functional form of the t distribution, which in turn is determined by

the number of degrees of freedom. The differences of the three curves in

each figure reflect the effect of the standard deviation on the probability

of vertical overlap. Another potential use of plots like these is that if a

t distribution with the "correct" degrees of freedom can be fitted to the

empirical probability distribution of the total vertical errors, then one can

use these curves to estimate probabilities of vertical overlap based on the

empirical distribution under different separation standards.

Probabilities of vertical overlap are plotted in Figure 14 for three, six,

and nine degrees of freedom, and in Figure 15 for 20, 30, and 40 degrees of

freedom, while the vertical dimension is set at 30 feet and standard deviation

at 100 feet. These figures illustrate the effects of the number of degrees

of freedom.

In Figure 16, the effect of aircraft vertical dimension on the

probability of vertical overlap under a t distribution is illustrated.
-32-

In Figure 17, probabilities of vertical overlap for three values (1000

feet, 1500 feet, and 2000 feet) of the separation standard are plotted

against degrees of freedom.

Table 2 provides a summary of computational results of probabilities of

vertical overlap under t distributions and three separation standards.
FAA

UERTICAL

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PROJECT

33

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STANDARDS

(FT)

,

T

DISTRIBUTION,

AIRCRAFT

H

=

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NU=20

SOLID I

SIGMA"

100,

DOT

(SIGMA*

150,

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SIGMA-200
FAA

VERTICAL

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PROJECT

- 34 -

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T

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FAA

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- 37 -

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SEPARATION

STANDARDS

(FT)

T

DISTRIBUTION,

Nlf=9,

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50

FT
FAA

UERTICAL

SEPARATION

PROJECT

- 38-

I

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DEGREES

OF

FREEDOM

DISTRIBUTION,

AIRCRAFT

H=30,

SIGI1A*100

FT

SOLID

!

S=

1000,

DOT!S=1S00,

DASHIS-2000

FT
ntf = 3 o S = 1000 S = 1500 S = 2000

100 0.41443-04 0.77663-05 0.24224-05

150 0.19458-03 0.27575-04 0.83643-05

200 0.69759-03 0.75358-04 0.20668-04

X z = 50 100 0.69497-04 0.12976-04 0.40428-05

150 0.32899-03 0.46085-04 0.13960-04

200 0.11686-02 0.12616-03 0.34499-04

nu = 6

A

z = 30 100 0.44697-05 0.22667-06 0.29225-07

150 0.87719-04 0.29478-05 0.35197-06

200 0.55314-03 0.23118-04 0.22065-05

A = 50

z 100 0.75476-05 0.38044-06 0.48891-07

150 0.14922-03 0.49529-05 0.58897-06

200 0.93220-03 0.38939-04 0.36942-05

nu = 9

A

z = 30 100 0.62310-06 0.86148-08 0.45941-09

150 0.41759-04 0.41075-06 0.19270-07

200 0.41774-03 0.83468-05 0.30685-06

>• N

II

CJ1 O 100 0.10766-05 0.14557-07 0.77145-09

150 0.71666-04 0.69563-06 0.32371-07

200 0.70729-03 0.14145-04 0.51604-06

Table 2: Probabilities of Vertical Overlap Under t Distributions

and Three Different Separation Standards

*nu is the number of degrees of freedom
-40-

nu = 20 o S = 1000 S = 1500 S = 2000

A

z = 30 100 0.79466-08 0.10053-11 0.20716-14

150 0.10748-04 0.50862-08 0.86057-11

* 200 0.26730-03 0. 12054-05 0.37601-08

A

z = 50 100 0.15037-07 0.17745-11 0.35626-14

150 0.18934-04 0.89814-08 0.14829-10

200 0.45667-03 0.20792-05 0.64758-08

nu = 30

A z = 30 100 0.11389-08 0.36517-14 0.44370-18

150 0.64429-05 0.71119-09 0.10170-12

200 0.23079-03 0.55640-06 0.52110-09

A

z = 50 100 0.23165-08 0.68603-14 0.71646-18

150 0.11512-04 0.13003-08 0.18147-12

200 0.39570-03 0.96967-06 0.91574-09

nu = 40

\

= 30 100 0.35899-09 0.63754-16 —

150 0.48489-05 0.22011-09 0.50713-14

200 0.21364-03 0.35694-06 0.16021-09

A-, = 50 100 0.76878-09 0.12920-15

150 0.87427-05 0.41289-09 0.93996-14

200 0.36703-03 0.62645-06 0.28580-09

Table 2 (continued)
-41-

5.3 The Normal /Double-Exponential Distribution

In addition to the standard deviation and the vertical dimension of aircraft,

two new factors appear in the normal /double-exponential probability distribution. They are p, which is defined as where and are the

standard deviations of the normal and the double exponential distributions;

and y, which is the weighting factor between the two probability distributions.

Computer runs were made where p was set equal to one, four, and seven and y

equal to 0.95 and 0.99. Values of the standard deviation for the normal distribution used in the computer runs were 100, 150, and 200 feet, consistent

with the standard deviations for the normal distribution used earlier. The

values of 6 for the double exponential distribution are obtained from the

values of p and a^. Values of cr^ and cr^ are computed in Table 3 to i 1 1 usstrate the effects of y and p on and a^£. Computational results are

plotted in Figure 18 to Figure 29, and a summary of the values of input parameters

for these figures is given in Table 4. Table 5 provides a summary of our

computational results on probabilities of vertical overlap under normal /doubleexponential distributions and three separation standards.

Since the normal /double-exponential distribution is the combination of a normal

distribution and a double exponential distribution, is generally less than

or equal to a^. The probability of vertical overlap is more sensitive to the

value of cr D£

than to the value of a^.
-42-

Y p [y + (l-y)p 2

] 2 a

N a DE

a NDE

0.95 1 1.0000 100 100 100.00

0.95 4 1.3229 100 400 132.29

0.95 7 1.8439 100 700 184.39

0.99 1 1.0000 100 100 100.00

0.99 4 1.0724 100 400 107.24

0.99 7 1.2166 100 700 121.66

Table 3: Values of a nr and a., nr. for

DE NDE

Certain Values of y, p, and
-43-

NORMAL/DOUBLE -EXPONENTIAL DISTRIBUTIONS

FIGURES Y P °N

18 0.95 30 1 100,150,200

19 0.95 30 4 100,150,200

20 0.95 30 7 100,150,200

21 0.95 50 1 100,150,200

22 0.95 50 4 100,150,200

23 0.95 50 7 100,150,200

24 0.99 30 1 100,150,200

25 0.99 30 4 100,150,200

26 0.99 30 7 100,150,200

27 0.99 50 1 100,150,200

28 0.99 50 4 100,150,200

29 0.99 50 7 100,150,200

i

1

Table 4: Values of Input Parameters

for Figure 18 to Figure 29
Y P A

z a

N

S = 1000 S = 1500 S = 2000

0.95 1 30 100 0.58099-09 0.48942-12 0.41568-15

0.95 1 30 150 0.16743-05 0.25404-09 0.22586-11

0.95 1 30 200 0.15094-03 0.68935-07 0.14219-09

0.95 1 50 100 0.10245-08 0.85940-12 0.72991-15

0.95 1 50 150 0.31325-05 0.43440-09 0.38538-11

0.95 1 50 200 0.26115-03 0.12450-06 0.24050-09

0.95 4 30 100 0.55425-06 0.94573-07 0.16145-07

0.95 4 30 150 0.32826-05 0.24601-06 0.75675-07

0.95 4 30 200 0.15367-03 0.51634-06 0.13831-06

0.95 4 50 100 0.92702-06 0.15815-06 0.26998-07

0.95 4 50 150 0.58321-05 0.41656-06 0.12631-06

0.95 4 50 200 0.26569-03 0.87462-06 0.23072-06

0.95 7 30 100 0.78841-06 0.28698-06 0.10451-06

0.95 7 30 150 0.34079-05 0.35022-06 0.17852-06

0.95 7 30 200 0.15370-03 0.54571-06 0.19696-06

0.95 7 50 100 0.13157-05 0.47882-06 0.17437-06

0.95 7 50 150 0.60418-05 0.58402-06 0.29768-06

0.95 7 50 200 0.26574-03 0.92436-06 0.32836-06

Table 5: Probabilities of Vertical Overlap

Under Normal/Double-Exponential Distributions

and Three Different Separation Standards
-45-

Y P A-z °N

S = 1000 S = 1500 S = 2000

0.99 1 30 100 0.12304-09 0.10194-12 0.86577-16

0.99 1 30 150 0.17802-05 0.54028-10 0.47053-12

0.99 1 30 200 0.16365-03 0.69219-07 0.30391-10

0.99 1 50 100 0.21933-09 0.17899-12 0.15202-15

0.99 1 50 150 0.33365-05 0.92954-10 0.80283-12

0.99 1 50 200 0.28315-03 0.12564-06 0.51627-10

0.99 4 30 100 0.11546-06 0.19701-07 0.33632-08

0.99 4 30 150 0.21153-05 0.51257-07 0.15767-07

0.99 4 30 200 0.16420-03 0.16245-06 0.28820-07

0.99 4 50 100 0.19312-06 0.32944-07 0.56240-08

0.99 4 50 150 0.38991-05 0.85562-07 0.26317-07

0.99 4 50 200 0.28406-03 0.28195-06 0.48077-07

0.99 7 30 100 0.16427-06 0.59792-07 0.21774-07

0.99 7 30 150 0.21414-05 0.72977-07 0.37199-07

0.99 7 30 200 0.16420-03 0.16858-06 0.41043-07

0.99 7 50 100 0.27413-06 0.99767-07 0.36330-07

0.99 7 50 150 0.39428-05 0.12170-06 0.62029-07

0.99 7 50 200 0.28407-03 0.29232-06 0.68427-07

Table 5 (continued)
FAA

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- 43-

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MORMAL/DOUBLE

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H=

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27
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-53-

6.0 Discussion and Conclusions

The phenomenon of decreasing probabilities of vertical overlap with

increasing separation standards, appears intuitively reasonable.

Probabilities of vertical overlap between two aircraft arise from the

overlapping region of the two probability distributions. Increasing

separation standards results in pulling these two probability distributions apart, thus reducing the overlapped area. Therefore, probabilities

of vertical overlap decrease as separation standards increase.

Of the three families of probability distributions that we have investigated, for a fixed value of standard deviation the probabilities of vertical

overlap under the assumption of a normal distribution decrease most

rapidly when vertical separation standards increase. For the t distribution, the probability of vertical overlap decreases- more rapidly for a

large number of degrees of freedom than for smaller numbers. For the

normal /double -exponential distribution, the probabilities of vertical

overlap initially decrease rapidly as the separation standard increases,

then change to an almost steady rate of decrease.

Vertical overlaps usually occur in the tail of the probability distribution of the total vertical errors. As can be seen in Figure 2, the

normal distribution has small short tails and the Cauchy distribution
-59-

has large long tails. The tails of the normal /double exponential

distribution and the t distribution lie in between these two extremes.

The shapes of the tails of these probability distributions in fact

determine the rate of decrease of the probability of vertical overlap •

as the separation standard increases. The above observations indicate

the importance of properly estimating the tail of the probability distribution of the total vertical error. Therefore, it is necessary in the

data collection plan that sufficient amounts of data be collected to

present a clear picture of the tail portion of the empirical probability

distribution of the total vertical errors. Attempts should be made to

fit a theoretical probability distribution to these data. Another advantage

of dealing with a theoretical probability distribution is that the probability of vertical overlap can be calculated with relative ease.

From the point of view of air traffic safety, the desirable probability

distributions for total vertical error of an aircraft would be normal

distributions with small standard deviations. The t distributions and the

normal/double exponential distributions have long tails which cause relatively

large probability of vertical overlap.

It is observed in Figure 9 that the curves of probabilities of vertical

overlap for normal distributions with standard deviations of 100 feet

and 200 feet cross each other when the two aircraft are separated

somewhere between 150 feet and 200 feet. A check by numerical methods

supports these computational results. An intuitive explanation of the

curves' crossing can be given as follows: For small (near-zero) values

of the separation S, overlap is associated mainly with the bodies of the

two aircrafts' error distributions, so that larger probabilities of overlap are associated with "large body" situations and thus with smaller
-60-

standard deviations. For larger values of the separation standard, the

overlap phenomenon is associated with the distribution-tails, and thus,

the liklihood of overlap is greater for larger tails, hence for larger

standard deviations. Macho! [8] also observed this; he observed that under

some circumstances, the better the navigation, the greater the probability

of collision, and he called this phenomenon "navigational paradox."

The effect of the vertical dimension of aircraft on the probability of

vertical overlap is visible using all three probability distributions

for the total vertical errors. In our limited computation of varying separation standards up to 2,000 feet, we observed that the minimum ratio of

probabilities of vertical overlap is approximately the ratio of the respective

aircraft vertical dimensions which was 50/30, and this happened at the

larger separation values (see Table 2 and Table 5). However, for those

distributions where the slope of the tail of the distribution is much greater

than zero, such as normal distribution, or for lesser separation values, the

ratio of probabilities of vertical overlap is generally higher than the ratio

of the respective aircraft vertical dimensions. It should be mentioned that

the sensitivity studies presented in this report are limited to the probability

of vertical overlap defined in Section 2. The effects of aircraft phugoid

and aircraft wake vortices were not considered.
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In this report, we have examined the effects of the functional form of

vertical error probability distribution, the parameters of the distribution, the aircraft vertical dimension, and the vertical separation standard

on the probability of vertical overlap between two aircraft. In Sections

2 and 3, the probability of vertical overlap was defined and the factors

affecting it discussed. A brief outline of computer programs developed

to compute the probability of vertical overlap was presented in Section 4,

and computational results along with their plots were presented in Section

5. Of the four factors that we have examined, the aircraft vertical

dimension is the least influential.

The functional form and the parameter such as the standard deviation of the

probability distribution of the total vertical error are critical to the

probability of vertical overlap. Once these two factors are known, the

vertical separation standard can be set at a level which assures that the

probability of vertical overlap between two aircraft is sufficiently small

to be acceptable to the air traffic conmunity.
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1. REFERENCES

Gilsinn, J. F. and Shier, D. R. , Mathematical Approaches to

Evaluating Aircraft Vertical Separation Standards, NBSIR 76-1067,

National Bureau of Standards, May 1976.

2.

Busch, A. C., Colamosca, B., and VanderVeer, J. R. , Collision

Risk and Economic Benefit Analysis of Composite Separation for

the Central East Pacific Track System, FAA-EM-775, Federal

Aviation Administration, June 1977.

3. Proceedings, Third Meeting of Panel for Review of the General

Concept of Separation, ICAO, RGCSP-WE/62, 1975.

4. Private Communication with Mr. Robert Rigolizzo of NAFEC, FAA.

5. ' IATA Report DOC- GEM/ 1951, Report on Vertical Separation over NAT

Region-1963, Published in 1964.

6. Commercial Aircraft of the World, Flight, p.1058, October 8, 1977.

7. Tables of Normal Probability Functions, Applied Mathematics

Series 23, National Bureau of Standards, 1953.

8. Machol , R. E., An Aircraft Collision Model, Management Science,

21_, p.1089, 10, June 1975.

9. Gilsinn, J. F., Problems in World-Wide Standardization of the Units

of Altitude Measurement, NBSIR 77-1386, National Bureau of Standards, February 1978.
NBS-114A (rev. »-78)

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO. NBSIR 80-1990

2.Gov't, Accession NoJ 3;. ffecifiteatfs Accession Noi

4. TITLE AND SUBTITLE

Probabilities of Vertical Overlap:

A Sensitivity Analysis

5. Publication Date MARCH 1980 ^ Performing Organization Code

.> \ : i,

; V: £

7. AUTHOR(S)

Howard K. Hung, Karla L. Hoffman

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

11. Contract/Grant No.

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP) 13. Type of Report 8< Period Covered

Federal Aviation Administration

Office of Systems Engineering Management

800 Independence Ave . , S. W., Washington, D. C, 20591

' ^

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15. SUPPLEMENTARY NOTES

I I Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or leas factual summary of moat significant information. If document includes a significant bibliography or

literature survey, mention it here.) Because of the potential increase in traffic at FL 290 and above, both current

and alternative vertical separation standards are being reviewed. A plan to

collect data on the vertical navigational performance of aircraft is also

contemplated. This report documents a sensitivity analysis carried out to

assess how different assumptions about the probability distribution of ’’total

vertical error” affect the probability of vertical overlap. The four factors

affecting the probability of vertical overlap which are examined in this study

are: the functional form of the vertical -error distribution function; the

standard deviation of this probability distribution; the vertical dimensions

of the aircraft; and the vertical separation standard. Probabilities of vertical

overlap were computed over a range of possibilities for each of these four

factors in order to discern the effect of each factor. A final section discusses

the findings of this study and draws some conclusions.

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name;

separated by semicolons)

Air safety; collision probability; probability distribution; sensitivity analysis;

separation standard; vertical error; vertical overlap

18. AVAILABILITY Unlimited

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