A purely stochastic Monte Carlo model is used to compare the relative midair collision course probabilities and mean closing velocities of four systems of rules for aircraft cruising altitudes as a function of altitude error:  (1) current U.S. federal rules, (2) random altitudes, and (3) two proposed alternatives to the current rules.  This model increments error while:  (1) counting collisions among cruising pairs of aircraft following the four rules being tested on random headings between randomly placed airports, and (2) calculating mean closing velocities for each rule.  The calculations verify that: (1) federal rules increase collision course probabilities by about four times more than for a chaotic system of aircraft cruising at randomly selected altitudes, (2) risk is directly proportional to the level of compliance, and (3) mean closing velocities resulting from the current rule are slightly less than for random altitudes, while being almost twice as high as for the proposed rules.  High closing velocities are shown to increase the collision probability.

Keywords:  Aircraft, midair, collision, rule, regulation

Federal Aviation Regulations (FARs) require pilots flying in level cruising flight to use altitudes that are a function of the magnetic course of the flight.  However, legal aircraft on mostly northerly and southerly courses have a high probability of being at the same altitude on head-on collision paths with the maximum possible closing velocity—and the resulting minimum possible probability for detecting and evading a collision.  The FARs being tested are presumably designed to reduce the probability of midair aircraft collisions.  This paper describes a purely stochastic Monte Carlo comparison of FAR effectiveness for reducing collisions in comparison to the random altitude selection rule and two proposed revisions to the FARs.

The survival-of-the-fittest philosophy suggests that the best pilots following a valid collision avoidance rule most faithfully should be rewarded with the lowest collision probability.  A sensitivity analysis is used to show that the current rules reward the most inaccurate piloting with the greatest probability of avoiding a collision.  This effect is similar to the previously described "navigation paradox."[1]  Alternatives to the current rules are proposed and tested to document that a change is possible to reward the best piloting with the safest flying.

This paper also discusses the discontinuous nature of the FARs at 3000 feet above ground level (AGL), and how the proposed rules are applicable at any altitude.  The proposed altitude rules suggest air traffic system modifications at some airports that could substantially reduce collision probabilities in airspace near airports where collision probabilities are significantly higher than the probabilities far from airports.[2]  Suggestions are also made for minimizing collision probabilities during periods of Air Traffic Control (ATC) equipment failure.

The FAR Sections 91.159 and 91.179 specify the altitudes to be flown by pilots following visual flight rules (VFR) and instrument flight rules (IFR), respectively.[3]  Section 91.159 applies at greater than 3000 ft AGL in the cases of both:  (1) VFR aircraft in any airspace, and (2) IFR aircraft only in controlled airspace operating with a "VFR conditions on-top" clearance in clear air.  Controlled airspace is mapped to show where ATC service is provided to all IFR flights, and to VFR flights according to the particular class of controlled airspace.  A VFR conditions on-top clearance is commonly used (by IFR flights only) in clear air to decrease the communication system load on both pilots and ATC where the pilot alone is responsible for separation under 91.113(b).  Section 91.113(b) holds both VFR and IFR pilots fully responsible for collision avoidance in clear air under the see-and-avoid rule.[4]  For both VFR and applicable IFR flights above 3000 ft AGL, 91.159 requires that westerly pilots (headings from 180 degrees through 359 degrees) fly at even integer thousands of feet plus 500 ft above mean sea level (MSL), while easterly pilots (headings from 0 degrees through 179 degrees) must be at odd integer thousands of feet plus 500 ft.  This paper calls this altitude formula "A3000."  Below 3000 ft AGL, aircraft may legally fly at randomly selected altitudes, unless specifically instructed otherwise by ATC.  This altitude formula is called "RAND."

Section 91.179, applicable only to IFR flights in mapped uncontrolled airspace (with no ATC service provided), requires that westerly courses are flown at even-integer-thousand-foot altitudes, while easterly courses are flown at odd-integer-thousand-foot altitudes.  In contrast, aircraft that are IFR in controlled airspace (outside the scope of 91.179), may fly at any altitude that is assigned or approved by ATC.

The first of the two new rule proposals for altitude selection being compared with the existing rules is called "RP2000."  With RP2000, aircraft cruising altitudes are selected in a continuous way across the entire available altitude range on 2000-ft intervals according to the formula:

Alt. = (heading degrees) × (2000 ft) / (360 degrees) + (2000 × n ft),                                             (1)

where n is an integer multiplier.  This multiplier is set to zero in the Monte Carlo model for the purpose of relative collision course probability and mean closing velocity calculations within a single 2000-ft interval.

While the RP2000 formula is simple enough to calculate, it lacks the human factors simplicity of the second rule proposed, "RP1000," which is as follows:

 Alt. = (heading degrees) × (1000 ft) / (360 degrees) + (1000 × n ft),                                             (2)

where n is an integer multiplier.  The proposed RP1000 is simpler to use in the cockpit because the pilot may determine the required altitude by making the degrees of magnetic heading (neglecting wind correction angle and local magnetic variation) on the magnetic compass or gyroscopic heading indicator match the degrees of rotation of the 100-ft hand of the altimeter.  With RP1000, the pilot at an initially random altitude is never more than 500 ft from a formula-permitted altitude.  In contrast, the A3000 and RP2000 formula altitudes are never more than 1000 ft from the random-altitude pilot needing to correct to compliance.  The reasoning behind creating the RP1000 formula in addition to the RP2000 formula is that an easier-to-follow rule may have better results in reducing the collision frequency than the less obvious RP2000, which is not so easily calculated from visual cues immediately available to the pilot.

4.1  Total Collision Probability
The probability of a collision between aircraft is influenced primarily by the three independent probabilities of:  (1) the aircraft being on a collision course (C), (2) the pilots failing to detect the collision course (D), and (3) the pilots failing to evade the detected collision course (E).  In general, a collision is only possible if C has occurred.  (One deterministic collision of aircraft that were not on a collision course occurred due to faulty and unnecessary evasive maneuvering.[5])  Given C, a collision is guaranteed if D occurs.  This event sequence has an end state probability of C times D.  Another collision end state occurs when, given C but not D, then E occurs.  This failure sequence has an end state probability of C times (1-D) times E.  The total collision probability is then the sum of the two end state probabilities:[6]

 COL = CD + CE(1-D)                                                                                                                 (3)

or,

 COL = C(D + E - DE).                                                                                                                 (4)

The normalizing technique divides both sides of Equation 4 for each of the various altitude selection rules by both sides of Equation 4 for the RAND rule.  This technique as applied to Equation 4 shows that if a relative performance difference between two rules influences C[i] /C[RAND], then that difference influences the total normalized collision probability comparison ratio, COL[i] /COL[RAND], by exactly the same ratio.

4.2  Probability of Collision Course
Mean free path (MFP) is the statistically determined mean distance that a particle will survive in a target volume without a collision.  The MFP in radiation science is proportional to the available material volume divided by the effective cross sectional area of the target atoms.[7]  The MFP of an airplane through a volume of airspace is similarly proportional to the flight volume divided by the vulnerable cross sectional area of the target aircraft.[8]  A lower MFP would then represent a higher collision probability.

Terrain and weather patterns may influence the available flight volume on a local level no matter what altitude selection rule is used.  Additionally, the proximity to airports also restricts the available flight volume (and biases the collision course convergence angles) through the concentrating influence of converging flight paths at runways.  For the purpose of relative collision course probability calculations in the enroute phase where the modeled rules apply, these local influences are assumed to be equal for all the altitude selection alternatives.  Because of the normalization process used in this paper, equal factors influencing the performance of the various altitude selection rules, divided by themselves, result in no influence on the results of relative collision course probability calculations.

4.3  Probability of Failing to Detect a Collision Course
The probability of failing to detect a collision course is influenced by many random variables such as the following:

(1) aircraft factors related to configuration, size, color, window clarity, and anticollision beacon power and coverage;
(2) environmental factors related to time of day, sky color, haze, and sunlight angle;
(3) pilot information management factors related to the individuals inclination to scan outside at periodic intervals depending on cockpit workload and distractions, in comparison to the finite time that an oncoming aircraft is visible before impact; and
(4) pilot health factors related to age, and time since last eye examination to adjust the vision correction prescription, in comparison to the rate of vision deterioration.

Obviously, considering image size alone, the pilot of a Piper Cub will probably see an oncoming 747 before the 747 pilot would see the Piper Cub.  However, the influential factors of No.s 1, 2, and 4, above, are assumed to be relatively equal for all the altitude selection rules considered.  Therefore, the probability of detecting an oncoming aircraft on a collision course is directly and linearly proportional to the fraction of time spent looking in the direction of the oncoming aircraft, and to the time that the image of an oncoming aircraft is large enough to detect.  Conversely, this probability is inversely and linearly proportional to the closing speed of the oncoming aircraft.  A higher closing speed results in a shorter time when the image of the oncoming aircraft is large enough to detect.
 The FAA-recommended technique for detecting oncoming aircraft is to scan a 10-degress arc along the horizon for at least one second before scanning the next 10-degree arc.  The eye can focus effectively on only a narrow viewing area.[9]  Assuming that aircraft structure and pilot human factors limit the scanning of the aft horizon, then the probability of looking effectively at an oncoming aircraft may be calculated.  For the 180 degrees of forward arc most likely to be scanned, there are 18, ten-degree arcs to scan for one second each.  Allowing two seconds to scan the instrument panel, there is only a 1/20th chance, or 0.05 probability, that a pilot will be effectively looking in the direction of an oncoming aircraft at any point in time.

For the hypothetical 20-second scan period (comparable to the measured results of one study[10]), if an oncoming aircraft is visible for at least 20 seconds, then collision course detection is obviously guaranteed.  However, high closing velocities, dirty or scratched windows, small aircraft size, haze, or sun angle can easily reduce the visible time of an oncoming aircraft to less than five seconds.[11]  In the realm of visibility durations that are less than the scan period, the probability of one pilot detecting an oncoming aircraft in the scanned arc is:

 1-D = (effective looking time + visible time) / (scan period duration)

OR,

1-D = 1,                                                                                            (5)

whichever is less.  Effective looking time is only the time spent looking in the 10-degree arc with the oncoming aircraft.  As an example, setting effective looking time at 1/20th of the scan period and visible time at 1/4th of the scan period (for a five-second closing), then the detection probability is only 0.3.  This hypothetical example shows that in high closing velocity encounters, the pilot has a mathematically poor probability of being responsible for his own collision avoidance, as required by FAR 91.113(b).[12]

The current altitude rules lower closing velocities only if both aircraft being considered are on nearly westerly or easterly courses.  With easterly and westerly courses, aircraft tend to be flying in the same direction at the same altitude such that no head-on, maximum velocity collisions are likely.  In contrast, aircraft on mostly northerly and southerly courses have a significant probability of being at the same altitude with head-on collision courses.  For example, closing courses of 181 and 359 degrees, both of which are westerly, could require two aircraft in the same 2000-ft interval to select the same even-thousand-plus-500-ft altitude, such as, 4500 ft.  Such perfectly legal courses give aircraft the highest possible closing velocities with the smallest possible probability for the pilots to detect the threat.

4.4  Probability of Failing to Evade a Detected Collision Course
The failure of the pilot to maneuver effectively away from a detected collision course is influenced by:  (1) aircraft maneuverability, and (2) pilot ability to maneuver in an appropriate direction to avoid a structural failure from either collision forces or maneuvering forces.  The maneuvering options are:  (1) a maximum acceleration pitch up, (2) a low or negative gravity pitch down, (3) a highly-banked steep turn (which could momentarily increase the effective collision cross section or collision probability), (4) an abrupt change in flight velocity, or (5) a combination of these maneuvers.  With these options, the time available from detection until impact directly and linearly controls evasion effectiveness.  It takes a finite amount of time for a change-in-direction maneuver to move one aircraft out of the path of the oncoming finite volume of the other.  The time available for maneuvering to safety is inversely and linearly proportional to the average component of the evasion velocity perpendicular to the closing path.  The average evasion velocity depends on pilot and aircraft maneuvering performance, which should be the same for any altitude selection rule.  The evasion time available is inversely and linearly proportional to the closing velocity, which is dependent on the closing velocity probability distribution as significantly influenced by the altitude rules modeled for this paper.

4.5  Independence of the Three Failure Events
It is necessary to demonstrate that the three failure events modeled in Equations 3 and 4 are independent.[13]  While it is easy to see that the collision course probability is unrelated to the detection and evasion probabilities, it is less obvious that the detection and evasion probabilities are independent of each other.  At first, pilot performance appears to be a common mode failure link between the detection and evasion failures.  However, the detection failure is controlled by the pilots visual scan effectiveness and time period in comparison to the length of time that a target is visible.  In contrast, the evasion failure is controlled by the pilots training and tolerance for maximum performance maneuvers or unusual attitudesand by the maneuverability of the aircraft.  Detection is a routine flight skill, while evasion is an emergency response skill.  Additionally, the success probability of evasion is directly proportional to the purely random time available between the moments of detection and impact.  This duration is random because it is controlled by how close the two aircraft are when a one-second look in the appropriate 10 degree arc actually detects the threat.  Considered in this light, there is sufficient justification for claiming that all three of the failures in Equations 3 and 4 are independent.

5.1  Random Course Generation
The universe of the model is a 10-by-10 matrix of evenly spaced, hypothetical airports.  From this matrix, two points are chosen at random using a uniform spatial distribution for aircraft departures and two points are chosen as destinations.  These four points define the random course headings that the hypothetical pilots use with the altitude rule being tested.

The reason for arbitrarily choosing the 10-by-10 matrix of airports is that a globally valid collision-reducing rule must be effective in any geography.  The aviation regulations of the USA are sometimes adopted by other countries hoping to incorporate the judgment of aviation safety leaders.  In the interest of presenting a calculation of global relevance, it is important to avoid modeling USA airport geography.  With the USA being generally longer in the east-west direction, a risk calculation based on USA geography would not be applicable to a country that is longer in the north-south direction.

5.2  Altitude Modeling
The FARs as applied between 3000 ft AGL and 29,000 ft MSL provide exactly four legal altitudes per 2000-ft interval.  However, only two of these altitudes per interval are legal for any one pilot at any one time.  Which two altitudes are legal depends on two factors:  (1) VFR or IFR operation, and (2) operation in controlled or uncontrolled airspace.  In general, no pilot is legally required to file a flight plan for VFR operations, while all IFR operations do require a flight plan.  A review of one pilots logbook with 350 total hours, including 50 cross country hours, showed that less than 20 hours of flight time was spent under any recorded flight plan.  With no universal flight plan requirement in the USA, it is impossible to use FAA flight plan records to calculate reliable unbiased fractions of aircraft that cruise at either the 91.159,  91.179, or random altitudes allowed below 3000 ft AGL.

Fortunately, the MFP formula allows bounding calculations to quantify the influence (above 3000 ft AGL) of 91.159 or 91.179 selection fractions on collision course probabilities across the full range of possible fractions of pilots using each altitude rule.  At the two extreme ends of the spectrum of possible fractions, one extreme has all pilots using only 91.159, while at the other extreme all pilots use 91.179.  In both of these cases, there are two legal altitudes used per 2000-ft interval.  At the midpoint of this spectrum of possible fractions, half of the pilots use 91.159 altitudes while the other half use 91.179 altitudes.  In this case, there are four legal altitudes per 2000-ft altitude interval with equal aircraft densities at all legal altitudes.

Actual collision data from one study indicate that the model of equal aircraft densities at both VFR and IFR altitudes is not realistic.  Of 74 enroute collisions studied, 57 were VFR-VFR collisions, 15 were VFR-IFR collisions, and 2 were IFR-IFR collisions.  In 37 of these collisions, both aircraft were in cruise mode, and in the other 37 cases, at least one aircraft was in cruise mode.[14]  These data indicate that it would be unreasonable to believe that there are equal fractions of VFR and IFR flights with equal densities at four legal altitudes per 2000-ft interval.  The data showing collisions dominated by the VFR-VFR category imply that the most valid point in the probability spectrum of possible VFR-IFR fractions is nearer to the all-VFR flight extreme.  This implication favors an assumption that the most statistically significant traffic should be using 91.159 altitudes, while traffic using 91.179 altitudes is statistically far less significant.

The data favoring a 100 percent VFR-only modeling fraction may be the result of navigation system design, training, and economic factors, all of which strongly bias IFR traffic into controlled airspace near busier airports and along Victor airways, where the 91.179 altitudes are not applicable and may not be used without an ATC clearance.  Additionally, IFR traffic in controlled airspace under ATC control is not randomly distributed.  Victor airways are four-mile-wide air routes generally structured between the busiest airports and the navigational transmitters used almost continuously by most IFR aircraft.  Some IFR aircraft are RNAV-, LORAN-, or GPS-equipped for direct flight using 91.179 through the uncontrolled airspace between Victor airways.  However, the minimum $5000 unit cost of such equipment is a financial bias favoring the use of controlled airspace with the significantly lower $1000 cost per VOR needed for controlled airspace navigation using the traditional transmitters that define Victor airways.  An additional bias is that IFR pilot certification typically includes no flight training or certification testing for skills with direct flight instruments.

Some may consider the assumption of a 100 percent VFR fraction to be erroneous.  To calculate the influence of this assumption being fully erroneous, the MFP formula is used to compare the A3000/RAND relative collision course probabilities at the all-VFR (or the all-IFR) extreme of the fractional probability spectrum with the 91.159-91.179/RAND relative probability at the equal-VFR-IFR midpoint of this spectrum.  While the FAR altitude formulas designate exact legal altitudes for flight, a pilot trying to maintain these altitudes is allowed latitude for instrument error, turbulence, and human error.  The human error allowed during flight testing for instrument pilot certification is ±100 ft.[15]  This allowable error, not counting instrument error, defines a 200-ft-thick legal slab of airspace about each FAR altitude.  For example, using a 200-ft span for cruising between 7000 and 9000 ft MSL, VFR pilots may fly in two altitude bands:  (1) 7400-7600 ft for easterly aircraft, and (2) 8400-8600 ft for westerly aircraft.  Recalling that MFP is proportional to volume, the all-VFR (or all-IFR) model has 400 ft of legal altitude per 2000-ft interval, with a resulting five times (from 2000/400) reduction in MFP compared to having no stratification of legal altitudes.  The equal-VFR-IFR model has 800 ft of legal altitude in four layers per 2000-ft interval, with a resulting 2.5 times (from 2000/800) reduction in MFP compared to having no altitude stratification.  Therefore, regardless of whether or not the all-VFR-fraction assumption is used for the MFP formula or for the Monte Carlo model, the collision course probabilities calculated using this assumption cannot be too high by more than a multiplier of two.  Actual collision data suggest that the all-VFR-fraction assumption is a better approximation of reality, which means that the most appropriate error factor is probably much closer to being a multiplier of one.  With either scenario, the general conclusions of this paper are similarly supported.

Having bounded the possible error in using the all-VFR-fraction assumption, it is now appropriate to describe the Monte Carlo model features for altitude modeling.  All of the aircraft cruising altitudes are modeled over a single 2000-ft altitude interval of 0 to 2000 ft.  While this is technically a violation of the FARs with respect to the 3000-ft AGL consideration, the model only counts collisions between the two aircraft positions following just one particular rule.  Both RAND aircraft are separately launched at their own unique altitudes according to the uniform random distribution as applied across the allowed altitude interval.  The A3000 aircraft are positioned at either 500 ft or 1500 ft according to whether their random courses are westerly or easterly, with the following modification.  The variable, Zslop, is used to define the span of allowed altitude error (due to piloting inaccuracy, instrument error, or turbulence) about the A3000, RP2000 and RP1000 altitudes.  The largest allowed error from the six formula altitudes is limited to half of Zslop.  Altitudes selected for RAND-following aircraft are not altered by Zslop because random error added to random altitudes is redundant randomness.

No attempt is made to determine the most appropriate probabilistic distribution for each of the random variables in the model, or how these distributions influence collision course probabilities.  In the interest of simplicity, uniform probability distributions are applied whenever random parameter variability is used in the Monte Carlo model.  The relative nature of normalized comparisons should neutralize any significant influence that a less than ideal probability distribution might have on the overall validity of the results from the Monte Carlo model.

5.3  Collision Counting
Having calculated the four altitudes for each of the two flight paths, the hypothetical aircraft positions are incremented from the departure airports to the destination airports.  Aircraft cruising velocities are randomly selected between 100 and 700 fps using a uniform distribution.  This range is chosen to approximate the velocity variability between a Piper Cub and a Boeing 747.  The range in aircraft velocity is varied because relative flight velocity influences the pilots probability of detecting and evading a collision course.  The faster of the two aircraft velocities is used for calculating time interval between tests for a collision event.  In the interest of counting all cases where collision courses exist, the model uses a distance to increment the faster aircraft along its flight path that is equal to the collision radius.  The collision radius is set at 50 ft to represent an approximately median distance (based on the Cub and 747) from the center of an aircraft to where an overlap between two aircraft counts as a collision.  At each increment along the two flight paths, the Pythagorean Theorem is used to calculate the four, three-dimensional separation distances between the four pairs of aircraft positions.  When the distance separating a pair of aircraft following one rule is less than the collision radius, the collision counter for that rule is incremented.

After the first of the two aircraft positions reaches the end of its flight path, the model repeats the above outlined process of:  (1) randomly selecting four new flight path end points, (2) calculating four new altitudes for each flight path based on the two new course directions and the four rules, and (3) incrementing both aircraft positions along their flight paths while testing for collision radius overlaps.  The model is programmed in Microsoft Qbasic for running under MS DOS 5.0 on a 33 MHz 386 Leading Edge Color Notebook Computer.  A technical report including a copy of the program is available from the author.[16]

Notice that the Equation 2 formula for RP1000 allows two pilots flying in the 0-2000 ft altitude range on the same magnetic course to be flying at either of two altitudes, depending on the selection of either zero or one for the integer n.  A true comparison of RP1000 performance with the other three rules that are modeled across the 2000-ft interval requires allowing the n of RP1000 to randomly be either zero or one.  In contrast, the Monte Carlo model calculates relative collision course probabilities as if the range of 1000-2000 ft is forbidden to pilots using RP1000, i.e., with n set to zero.  Truth table modeling for two aircraft flying to RP1000 in one 2000-ft altitude interval at random values for n of zero or one shows that in one half of the cases, the two aircraft are in different halves of the 2000-ft interval with zero probability of being on a collision courseas long as the error from the formula altitude is less than ±500 ft (or Zslop < 1000 ft).  Therefore, RP1000 collision courses counted in the model for the 1000-ft interval are divided in half to produce an equivalent probability for comparison with the other 2000-ft interval rules.

5.4  Closing Velocity Means
Whenever the model detects a collision event, the orthogonal velocity components of both colliding aircraft are used to compute the closing velocity of that collision.  Each closing velocity is then added to the velocity accumulator for the altitude rule used by the colliding aircraft.  At the end of the collision test series for each set of input variables, the values in the four velocity accumulators are divided by the number of collisions for each altitude rule to find the four respective mean closing velocities.

5.5  Sensitivity Analysis for Altitude Precision
The model is used to test the four altitude rules to compare their relative collision course probabilities and closing velocities as a function of the error in maintaining the rule-required altitude.  The model variable for altitude error, Zslop, is incremented from 0 to 1000 ft in 50-ft steps.  At each value of Zslop, 3000 pairs of random courses are tested to count collisions.  This process is repeated seven times to allow a more meaningful calculation of the one-sigma uncertainties of mean collision count ratios and mean closing velocities.

6.1  Collision Course Relative Probabilities
For relative collision probability comparisons, all Monte Carlo collision counts for each increment of Zslop are divided by respective RAND counts as a way of computing normalized, relative results.  These results, along with similarly calculated results from the MFP formula, are presented for comparisons in Figure 1.  The results show that zero altitude error by precisely piloted aircraft following the FARs results in a collision course probability more than six times higher than for aircraft at random altitudes.  At a Zslop of 200 ft, the value of about four for the A3000/RAND collision course ratio is comparable to the value of five calculated earlier using the MFP formula.  While the A3000/RAND Monte Carlo and MFP comparison data follow the same general curve shape, smaller values of Zslop cause MFP values to be increasingly higher than those from the Monte Carlo ratios.  This trend is tied to the MFP A3000/RAND collision count ratio approaching infinity at Zslop = 0.  In contrast to the A3000 collision counts being significantly higher than the unity level of RAND, Figure 1 shows that the RP2000 and RP1000 collision counts are significantly less than the RAND collision counts.  In comparison to the A3000 formula with a Zslop of 200 ft, the RP2000 and RP1000 collision counts are about 4.6 times lower (3.92/0.86).
 
 
6.2  Mean Closing Velocity Results
Figure 2 shows the influence of altitude error on the visible time of oncoming aircraft within an assumed visible range of three miles.  The three-mile visible range was chosen based on:  (1) comments by Jefferson County Airport (BJC) control tower operators about their typical visual detection range under good conditions for the smaller VFR aircraft typically involved in midair collisions, and (2) a typical minimum visibility requirement for VFR flight in most categories of airspace.  The closing velocity influence is important because if two aircraft are on a collision course while flying in approximately the same direction, and if their flight velocities are about the same, then the time to collision could be so long that one of the aircraft may reach its destination before the collision actually occurs.  Additionally, if two aircraft are on a collision course going in about the same direction at nearly the same speed, then two other cases are possible.  In the first case, a slightly faster aircraft in front would pull away from the trailing aircraft.  In the second case, a slightly slower aircraft in front could be approached in a very gradual manner by the overtaking aircraft, giving the overtaking pilot a relatively large amount of time to impact for noticing the oncoming aft view of the lead aircraft.
 
Figure 2 shows that A3000 does create a slight improvement in closing velocity safety compared to RAND, where A3000 has about 1.2 times the three-mile visible range closing times of RAND.  This slight improvement is reasonable considering that nearly westerly and easterly courses do sort aircraft following A3000 into same-direction altitudes (while northerly and southerly courses are not being so beneficially sorted).  In contrast,  RP2000 and RP1000 have visible closing times that are about 2.1 times greater than the visible times from RAND at zero error.  This significant difference from the results of comparing A3000 to RAND is reasonable because RP2000 and RP1000 do not allow any altitude assignments to create head-on collision situations, as does A3000 for mostly northerly and southerly courses.

While increasing altitude error has no substantial influence on degrading the closing velocity performance of A3000, Figure 2 shows that increasing the error did create a significantly decreasing visible time trend for both RP2000 and RP1000.  The RP2000 violations must be about ±500 ft in error (with a Zslop of 1000 ft) before visible time performance has degraded to the visible-time-reducing protection offered by the currently mandated A3000.

6.3  Relative Collision Probability Calculation Considerations
Attempts were made to incorporate the relative collision count ratios and the relative closing velocities into Equation 4 to calculate the relative collision frequency ratios applicable to the four tested rules.  However, of the three terms influenced by velocity, two have a positive sign and one, DE, has a negative sign.  Therefore, it would not be strictly accurate to suggest that doubling the closing velocity directly doubles the total relative contribution of closing velocity to the total collision risk.  This suggestion is only approximately valid if D and E are both significantly less that one.  For example, if D and E are on the order of 0.01 and they are both doubled, then the D + E part of Equation 4 is also doubled, while the negative DE term is too small in magnitude to appreciably diminish the doubling from the two positive terms.

Another complication of including relative velocity effects in calculations of relative collision probability is that doubling the closing velocity increases D in a nonlinear way, as controlled by Equation 5.  Doubling the closing velocity may multiply D by anywhere from 1.1 (in the range of two-second visible times) to 10 (in the range of visible times slightly less than the scan period duration), depending on the size of the variables in Equation 5.

Ideally, when the time to impact is greater than the visual scan period, collision course detection is guaranteed.  Notice in Figure 2 that with Zslop = 0, the time to impact for RP2000 and RP1000 is almost twice that of A3000 and RAND.  If the hypothetically optimized 20-second scan period from the Equation 5 example is doubled (for example, by a two-second scan per 10-degree arc, 28-second visible time from Figure 2, and a four-second panel scan), then A3000 has a detection probability of only 0.75, while both RP2000 and RP1000 detection probabilities would still be 1.0 due to their 50-second visible time.  The same degradation of the A3000 detection probability, starting with the original Equation 5 example values, occurs if the three-mile detectability is halved by haze, solar glare, or small aircraft size (while RP1000 and RP2000 detection probabilities would still be 1.0).

In any case, the mathematical complications with the closing velocity terms in no way compromise the validity of the Equation 4 influence of normalized collision count ratios on total collision probability ratios.  Doubling the probability of being on a collision course certainly doubles the total probability of having a collision.  The only deficiency caused by this paper not calculating velocity effects on Equation 4 is as follows.  In a comparison of, for example, A3000 to RP1000, if A3000 has higher collision count probabilities and higher closing velocities than RP1000, then the collision course ratios shown in Figure 1 will underestimate the benefit from abandoning A3000 in favor of RP1000.  Only in the case of one rule having higher collision counts (more risk) and lower closing velocities (less risk) than another would the closing velocity contribution to total risk have to be fully calculated to judge which rule provides for the better collision-reducing performance.

Figures 1 and 2 show that this latter case does exist for comparing A3000 to RAND.  However, the relative size of the less safe, greater A3000 collision count ratio (about four to six times greater than RAND for typical altitude deviations) significantly overwhelms the relative size of its more safe, lower mean closing velocity ratio (about 0.85 of RAND).  The size of the overwhelming less safe collision count ratio compared to the slightly more safe closing velocity ratio is sufficient basis for believing that A3000 overall produces more collision risk than RAND.  This reasoning correlates with others findings that head-on traffic, which is allowed by A3000, is almost 30 times as risky as same direction traffic.[17]

6.4  Comparison of Calculated Results to Actual Data of Midair Collisions
An unsuccessful attempt was made to compare the results of the Monte Carlo model to actual data of midair collision rates for cruising aircraft.  A significant complication is that actual collision events used for this comparison must first be sorted into the following categories:  (1) both aircraft are following FAR cruising altitude rules, (2) one aircraft is following the FAR rules and the other is flying at a random altitude, and (3) both aircraft are flying at random altitudes.  This sorting is mandatory because the Monte Carlo model prohibits all collisions in the second category.

Unfortunately, actual collision data is impossible to accurately sort into these categories.  In the first case, a high percentage of collisions have no known altitude.  One study of 52 events contained 11 at unknown altitudes, for a possible categorization error of 21 percent.  In the second case, collision altitude data that are available are rounded to the nearest 100 ft.[18]  The rounded altitudes on and nearest the 91.159 altitudes then comprise 3/10ths of all possible reported collision altitudes.  However, a RAND aircraft has a 30 percent probability of being at or adjacent to a 91.159 altitude such that its collision can be incorrectly categorized in a 91.159 collision.  This error contribution could be as high as 30 percent.  The coincidence of RAND collisions at 91.159 altitudes makes both the numerator and denominator of the 91.159/RAND actual-data relative collision frequency ratios vulnerable to 30 percent error for comparisons to the Monte Carlo A3000/RAND ratios.  In the third case, without altitude versus time data prior to impact, it is impossible to objectively tell if any specific collision participants were actually obeying 91.159 or RAND.  Section 91.609(c) even now only requires flight data recorders on aircraft with more than 10 passenger seats.[19]  At the time of the above-referenced study, altitude-reporting transponders were rarely used in most general aviation aircraftmeaning that even ATC flight-tracking computer records would have been incomplete for altitude versus time data.  Additionally, survivor testimony is highly subjective.  Given the USA Fifth Amendment allowance against self incrimination (and the severe liability implications of violating an FAR and causing a collision), survivor recollections of compliance to 91.159 requirements are likely to have a very high bias error.

This paper illustrates a case where Monte Carlo modeling gives more accurate probability calculations than any generated from actual data that cannot be reliably categorized.  The difficulty with measuring collision altitude is acknowledged implicitly by the lack of a collision altitude data field on the current form used by National Transportation Safety Board accident investigators, Factual Report Aviation Accident/Incident Form 6120.4 Supplement OIn-Flight Collision.  The use of MFP calculations is necessary for the independent validation of the Monte Carlo results, as shown in Figure 1.  The combined use of the Monte Carlo and MFP models makes the lack of a comparison to actual (but unsortable) data acceptable by:  (1) focusing on relative collision course probability comparisons among the four altitude rule options, and (2) bounding the worst-case possible error for the hypothetically faulty all-VFR assumption.

Studies of actual collision data found the dominant closing geometry to have faster aircraft colliding with the rear of slower aircraft.  Collisions generally occur in the traffic pattern, primarily on final approach.[20]  These two generalities are statistically coupled to each other because:  (1)  airport traffic has the highest density of all, and (2) airport traffic patterns systematically force most airplanes to fly in the same (nonrandom) directions such that head-on collisions are less frequent.  It is reasonable that airport area statistics overwhelm the data from random cruise-mode collisions.  Unfortunately, the biased data near airports has predictable effects that are not relevant to the cruise-limited model in this paper.

The FAR permission to ignore the altitude-from-heading formula below 3000 ft AGL is inconsistent.  This inconsistency allows a RAND anarchistic environment below the 3000-ft level with an A3000 structured environment immediately above.  Additionally, the vast majority of aircraft without radar altimeters have no accurate means of measuring AGL altitudes in areas where surface elevations change continuously.  There is no reasonable, systematic way to justify that collision probabilities driven by geometric relationships suddenly are altered by distance from the ground such that an altitude versus heading structure is good in one volume while chaos is good in a similar, adjacent volume.  In contrast to the existing FAR, the proposed RP1000 is mathematically applicable all the way down to safe terrain-clearing altitudes.

Another advantage of RP1000, in the human factors domain, is that a maximum of only a 500-ft climb or descent is required to achieve conformance to the rule after leaving an airport at any random terrain-clearing altitude.  Both A3000 and RP2000 have a maximum of a 1000-ft climb or descent to reach a required altitude.  Generally, the easier it is for humans to comply with a rule, the more often they will do so.

Additionally, RP1000 is theoretically capable of providing a single runway airport with a traffic pattern that is defined as a continuously curving path that would minimize collision probabilities throughout the entire arc for approach and departure.  This idea would require a pilot approaching the runway to descend along a curve that continuously maintains a magnetic heading measurement approximately equal to the angular degrees of the 100-ft hand of the altimeter.  With this approach, even relatively low-altitude, sightseeing pilots out for a short tour of the airport vicinity in proximity to landing and climbing aircraft on an RP1000 curve would benefit from such a collision-probability-reducing altitude rule.  Such a curved pathor for that matter, a straight path in cruisewould place conflicting traffic pattern threats into a narrow convergence angle such that a very short scan period of as little as five seconds focused within 30 degrees of the flight path would allow for a greatly reduced value of D from Equation 5.

While positive control of aircraft flight paths as administered by ATC effectively prevents aircraft from colliding, its effectiveness is limited to those aircraft being controlled in airspace where all aircraft are controlled at all times.  Recently, after ATC system failures, aircraft have reported being uncomfortably close to other unexpected air traffic.  It is especially important that during system failures with no positive control by ATC, that uncontrolled aircraft fly at altitudes that are mathematically consistent with the natural order of geometric factors affecting collision probability.  Very high altitude aircraft flying under RP1000 while under ATC control would still have a valid collision avoidance strategy for periods of temporary difficulties with ATC equipment.

Considering high altitude traffic, all flights above 18,000 ft are required by FAR 91.121 to use the same barometric pressure setting of 29.92" Hg.[21]  As with the present rule, the proposed rules would have all aircraft on a certain heading slouching up or down together when flying from one barometric pressure zone to another.  Considering allowed instrument error per FAR 43, Appendix E, Table 1, lower pressures at higher altitudes have greater instrument error than at lower altitudes and greater pressures.  For example, the allowed tolerance at 6000 ft is ±40 ft, while at 40,000 ft the tolerance is ±230 ft.[22]  Using the square root of the sum of the squares of error formula required for combining independent inputs of altimeter and pilot error,[23] the 40,000-ft instrument error with the 100-ft maximum pilot error gives a total of ±251 ft (or Zslop = 502 ft).  Figure 1 shows that high altitude aircraft following A3000 at Zslop = 502 ft are 2.8 times more likely (1.67/0.60) to be on a collision course than aircraft following RP1000but only for 2000-ft altitude intervals.  For aircraft above 29,000 ft, the actual altitude increment is 4000 ft for two legal cruising altitudes.  Using the MFP formula to adjust the high altitude survival probability from the modeled 200/2000 factor to the actual 200/4000 factor (noting that aircraft sizes are constants), the current high altitude rules are 5.6 times more likely to cause a collision than RP1000 at 40,000 ft.  This calculation, being based on purely random situations, is only applicable at high altitudes during periods of ATC equipment failure of long enough duration for the nonrandom, structured, spatial intervals between planes under positive control to degrade into more randomized separation intervals.  In contrast, uncontrolled VFR traffic at lower altitudes follows the randomized separation scheme nearly all the time.

The current FARs 91.159 and 91.179, on the average, provide negative benefit for reducing aircraft collisions, and should be abandoned.  The idealized system filled with skillful pilots strictly adhering to FARs at required altitudes has a significantly greater probability of creating collision courses in comparison to even an anarchistic system of sloppy pilots flying at random altitudes.  Additionally, these FARs encourage closing velocities that are nearly as high as those achieved by a system of pilots flying at random altitudes.  The FARs are mathematically driven toward providing a risk of collision that is directly proportional to the level of compliance for two reasons:  (1) they increase the concentration of aircraft in higher density, narrow altitude bands; and (2) they systematically allow head-on collisions with nearly the maximum possible closing velocity and the minimum possible collision course detection and evasion probabilities.  Equation 5 allows the evaluation of situations in which pilots are statistically incapable of avoiding a collision as required under the see-and-avoid rule of 91.113(b).

Figure 1 shows that A3000 violates the survival-of-the-fittest philosophy for collision course probability, which is relatively insensitive to RP2000 and RP1000 altitude errors.  Figure 2 shows that RP2000 and RP1000 do follow the survival-of-the-fittest philosophy for closing velocity, which is almost insensitive to A3000 altitude errors.  The RP2000 performance is better than RP1000 for closing velocity.  However, the structuring of RP1000 around the typical cockpit presentation of heading and altitude information provides a better human-factored alternative for lowering collision probability.  Finally, RP1000 could be adopted for all airspace, even at high altitudes, to provide a greatly increased mean free path during ATC system failures.

[1]Machol, Robert E., "An Aircraft Collision Model," Management Science, Vol. 21, No. 10, June 1975, 1094, 1101.

[2]How to Avoid a Midair Collision, U.S. Dept. of Transportation, FAA, FAA-P-8740-51, 1-2.

[3]FAR/AIM 96 (Englewood, Colorado:  Jeppesen Sanderson, Inc., 1996), F-158-159, F-163-164.

[4]FAR/AIM 96, F-146.

[5]How to Avoid a Midair Collision, U.S. Dept. of Transportation, FAA, FAA-P-8740-51, 5.

[6]Mendenhall, William and Terry Sincich, Statistics for Engineering and the Sciences:  Third Edition (San Francisco:  Dellen Publishing Company, 1992), 107.

[7]Foster, Arthur R. and Robert L. Wright, Jr., Basic Nuclear Engineering:  Second Edition (Boston:  Allyn and Bacon, Inc., 1973), 174-175.

[8]Machol, Robert E., "Effectiveness of Air Traffic Control System," Journal of the Operational Research Society, Vol. 30, No. 2, 1979, 116.

[9]FAR/AIM 96, A-208.

[10]How to Avoid a Midair Collision, U.S. Dept. of Transportation, FAA, FAA-P-8740-51, 8.

[11]Machol, Robert E., "An Aircraft Collision Model," Management Science, Vol. 21, No. 10, June 1975, 1095.

[12]FAR/AIM 96, F-146.

[13]Mendenhall, 95.

[14]Machol, Robert E., "Effectiveness of the Air Traffic Control System," Journal of the Operational Research Society, Vol. 30, No. 2, 1979, 114-115.

[15]Instrument Rating For Airplane and Helicopter Practical Test Standards FAA-S-8081-4A (Renton, Washington:  Aviation Supplies and Academics, Inc.), 1-8.

[16]Patlovany, Robert, A Program for Comparing Aircraft Collision Rates and Closing Velocities for Different Altitude Selection Rules, unpublished.

[17]Machol, Robert E., "An Aircraft Collision Model," Management Science, Vol. 21, No. 10, June 1975, 1094.

[18]Machol, Robert E., "Effectiveness of Air Traffic Control System," Journal of the Operational Research Society, Vol. 30, No. 2, 1979, 117.

[19]FAR/AIM 96, F-199.

[20]How to Avoid a Midair Collision, U.S. Dept. of Transportation, FAA, FAA-P-8740-51, 1-2.

[21]FAR/AIM 96, F-147.

[22]FAR/AIM 96, F-37.

[23]Holman, J.P., Experimental Methods for Engineers:  Third Edition, (New York:  McGraw-Hill Book Co., 1978), 45.

As a pilot, I usually fly a Cessna 172.  It has a wingspan of 36 ft, a length of 25.5 ft, and a height of 8.75 ft.  I also built a KR-2.  It has a wingspan of 21 ft, a length of 14.5 ft, and a height of about 4.5 ft.  Using the square root of the sum of the squares of half dimensions to approximate an effective collision radius for these two aircraft, I calculate that the Cessna 172 has a collision radius of about 22.5 ft, and the KR-2 has a collision radius of about 13.0 ft.  As compared to the significantly larger 50-ft collision radius used in the above published report, I wondered how Figures 1 and 2 would change for a more realistic collision radius of 25 ft.  Additionally, in the above published article, I modeled a random flight velocity variability of 100-700 fps.  Since it is a violation of FAR 91.117 to fly faster than 250 knots below 10,000 ft, and it is likewise illegal to fly faster than 200 knots below 2500 ft near a Class C or D airport, I also wondered how a more reasonable flight velocity range of 100-300 fps would affect Figures 1 and 2.  Below are simplified versions of how using a less conservative, more realistic collision radius and velocity variability would affect Figures 1 and 2.  I reran the Monte Carlo collision modeling program with only these two variable changed.  Additionally, since I now have Pentium that is 30 times faster than the 386 used for the previous calculations, I counted 250,000 pairs of aircraft for each rule at each incremented value of altitude error.  With the greater number of pairs tested, I decided to leave off the one-sigma boundary markings on the new charts.  Additionally, the RP2000 data was not presented, because it is not a practical rule to follow from the perspective of cockpit human factors considerations.  I also change the Figure 2 axis to simply show data for mean collision closing velocity.

The New Figures 1 and 2, below, compared to the Figures 1 and 2 in the above published article, show that the smaller the aircraft, the greater the relative risk unnecessarily cause by FARs 91.159 and 91.179.  Additionally, the Cessna 172-sized aircraft have over 10 times more risk when complying with the regulations, compared to the risk for ignoring the regulations at random.  It is easy to extrapolate that for KR-2 pilots the risk of collision with strict compliance is probably 20 times greater than random noncompliance.  In any case, strict compliance with RP1000 still halves the closing velocity and the visible time to impact achieved with the FARs.