J.W. Andrews

Modeling of Air-to-AirVisual Acquisition

A mathematical model of air-to-air visual acquisition has been developed and validatedin a series offlight tests atLincoln Laboratory. The model describes the visual acquisitionprocess as a nonhomogeneous Poisson process in which the probability of visualacquisition per unit of time is proportional to the solid angle subtended by the target.The model has proven useful in the investigation of actual midair collisions, as well asin the evaluation of collision avoidance systems.

The Cerritos Midair Collision

August 31, 1986 was a typical warm, sunnysummer day in southern California. At 11 a.m.,three people boarded a single-engine PiperArcher airplane at the Torrance, Calif., airfieldand departed on a flight to the resort area ofBigBear, Calif. The pilot of the Piper was WilliamKramer, a 53-year-old metallurgist who ownedthe aircraft. On board were his daughter and hiswife. The Kramers planned a day ofvacation atBig Bear and a return flight to Torrance laterthat evening. As the Piper was climbingout fromTorrance, Aeromexico flight 498, a DC-9 air­craft, was descending into Los Angeles on thelast leg of a regularly scheduled flight that hadoriginated in Mexico City. In the radar controlfacility at Los Angeles International Airport, airtraffic controllerWalterWhite cleared Aeromex­ico to descend for the approach. No one seemedto suspect that anything was amiss until, at6,500 feet, the Piper collided with the tail oftheDC-9 (Fig. 1). The Kramers were fatally injuredin the collision; the Piper spunoutofcontrol andslammed into a school yard in the suburbantown of Cerritos. The DC-9 plunged to earth ina residential area, killing all 64 persons on boardtogether with 15 people on the ground (Fig. 2).

In the months that followed, investigatorsfrom the National Transportation Safety Board(NTSB) carefully reconstructed the events thatled to the tragedy. The wreckage was reas­sembled, measured, and photographed. Thecockpitvoice recorderwas recovered and pains-

The Lincoln Laboratory Journal Volume 2, Number 3 (1989)

takingly transcribed at the NTSB laboratory inWashington, D.C. Air traffic personnel, airlineofficials, eyewitnesses, friends, and relatives ofthe pilots were interviewed. Among the manyimportant questions to answer was a familiarone to the NTSB investigators: why didn't thetwo aircraft see each other in time to avoid acollision?

In pursuit of the visual acquisition question,NTSB investigator Malcolm Brenner contactedLincoln Laboratory about studies conducted aspart of Lincoln Laboratory's collision avoidancework. Lincoln Laboratory had developed amathematical model of visual acquisition andhadjust completed a series offlight tests directlyapplicable to Cerritos. Lincoln agreed to supportthe NTSB, and the details of the Cerritos acci­dent were soon being analyzed with the Lincolnmodel.

Origin of the Model

Visual acquisition is not a deterministic pro­cess; iUs described by models that are probabil­istic in nature. For small targets, acqUisitiontakes place onlywhen the direction ofthe pilot'svisual search is aligned almost exactly with theline ofsight to the target. The randomness ofthesearch process, combined with the randomnessof the detection probability at a given angularoffset, causes acquisition times to vary greatly,even under identical visual conditions. An air­craft on a collision course is usually well abovethe visual resolution threshold of the eye before

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Andrews - Modeling ojAir-to-Air Visual Acquisition

Fig. 1-lmpactgeometry for the Cerritos, Calif., midair collision between an Aeromexico DC-9 and aPiper Archer, on August31, 1986. According to eyewitnesses, neitheraircraft took any evasive actionbefore impact.

it is acquired. Pilots have described the phe­nomenon ofacquisition well above the thresholdby saying that targets seem to "pop up out of thesky." In reality, the targets are visible for sometime before the search results in acquisition.

The Lincoln Laboratory model originateddUring testing ofa ground-based collision avoid­ance system known as the Automatic TrafficAdvisory and Collision Avoidance System. Dur­ing the flight tests, subject pilots flew near­collision encounters in two Lincoln Laboratoryaircraft based at L.G. Hanscom Field in Bedford,Mass. Radar measured the paths of the aircraft,and radio transmission determined the timewhen the pilots visually acquired each other. Bycorrelating the speeds and distances with thetimes of visual acquisition, the flight tests pro­duced quantitative measures of pilot visualacquisition perfonnance.

476

The immediate question to be answered wasthe degree of benefit from the automatic trafficadvisories provided by the collision avoidanceequipment. At first glance, it might seem thatthe benefits could be characterized by somesimple statistic such as the range at which theapproaching traffic was seen. But two difficul­ties arose. The first problem was that the visualacquisition process was highly variable andthus required a suitable statistical approach.The second and more significant problem wasthat many factors affected the difficulty ofvisualacquisition. Among these factors were the speedand geometry of approach, the size of the targetaircraft, the timing and accuracy of traffic ad­visories, and the preVailing atmospheric visibil­ity (visual range). How could visual acqUisitiondata collected at one set of speeds against asingle aircraft be applied to the general problem

The Lincoln Laboratory Journal. Volume 2. Number 3 (1989)

Fig. 2-Aftermath of the Cerritos, Calif., midair collision.Fifteen people on the ground were killed in addition to 64people on the airplane. The accident occurred on a clearday when the reported visual range was 14 nmi.

involving a range of speeds and a variety ofaircraft? If the conclusions were to be generallyapplicable. it would be necessary to devise ameans of extrapolating experimental results toother conditions. The mathematical model ofvisual acquisition was developed to serve thispurpose.

The first step in developing the model was toestablish a mathematical formalism that couldserve as a powerful yet general basis for analy­sis. The key to such a formalism proved to be theuse ofthe instantaneous visual acquisition rate.At any given instant. the visual acquisition rateAcan be defined as follows:

?c(t) = lim P(acq in M) . (1)M---70 tlt

This rate is the probability of visual acquisi­tion per instant time. IfA = O.02/s. for example,then the probability of visual acquisition in onesecond of search is approximately 2%. Thecumulative probability of visual acquisition isobtained by evaluating the acquisition proba­bilities for each instant as the target aircraftapproaches. If Awere constant. then the resultwould be a set of simple equations that corre­spond to the classic Poisson process. a processused widely to describe physical processes suchas the rate of fission of radioactive isotopes.

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Andrews - Modeling ojAir-to-Air Visual Acquisition

Unfortunately, the air-to-air visual acquisitionprocess is complicated because Ais not constantwith time. In fact. A increases steadily as twoaircraft on a collision course approach closertogether. The cumulative probability of visualacqUisition must then be written

P(acq by t,) = 1- exp[-IA(tldtl (2)

This equation describes a nonhomogeneousPoisson process and provides a general mathe­matical formulation that can be used to describemany different visual search situations.

To apply Eq. 2. an expression for Amust befound. Several sets oftest datawere examined inaddition to the Lincoln Laboratory data. Theother sets of data included a classic flight testconducted in DC-3s by Wayne Howell for theCivil Aeronautics Administration in 1957 [2],and data collected by Control Data CorporationdUring a series of air-to-air photographic mis­sions in 1973 [3). When properly analyzed, alldata sets supported the conclusion that thevisual acquisition rate is proportional to theangular size of the visual target. That is,

(3)

where A is the visual area presented by thetarget aircraft, ris the range to the target. and f3is a constant to be determined.

The above expression is suitable only whenthe atmosphere is sufficiently clear at the rangesof interest. In fog or haze. the contrast of targetscan decrease rapidly with range; this depend­ency must be included if f3 is to be modeled as aconstant. Laboratory experiments show that thedetectability of a static target is actually deter­mined by the productofthe target size and targetcontrast. This recognition suggests that thesize-contrast product should be substituted forthe size expression in Eq. 3. A relationshipknown as Koschmieder's law states that. in ahomogeneous atmosphere, apparent contrastdegrades exponentially with visual range. Thus.if Co is the inherent contrast of the target withits background, then at range r the apparent

477

Andrews - Modeling ojAir-to-Air Visual Acquisition

contrast is reduced to

Co exp( -2.996 ~).

The constant -2.996 arises from the defini­tion of visual range as the range at whichapparent contrast is reduced to 5% of the inher­ent contrast.

The resulting expression for A. is

A = f3 ~ exp ( -2.996 ~) (4)

where A is the visual area presented by thetarget aircraft, ris the range of the target, and Ris the visual range. Here the inherent contrast isincluded in the value of (3.

Numerical integration is usually required tofind the cumulative probability of visual acqui­sition given by Eq. 2. However, in certain specialcases a' simple closed-form solution can beobtained. For example, consider the commoncase in which the following conditions apply:(a) the visual range is so large that it plays norole in degrading contrast, (b) the target aircraftis on an unaccelerated collision course, and (e)the value of (3 and A are constant. Condition aallows us to set R to infinity. Condition b allowsus to write the range to the target as r =- it where ;- is the closing rate and t is the timeto collision. Condition e allows the integration tobe carried out with the following result:

P(acq by tz ) = 1 - exp( .~A J. (5)r tz

The Value of Traffic Advisories

Air traffic controllers assist visual acquisitionby providing traffic advisories for nearby aircraftdetected by radar. Such an advisory might be"American 578, you have traffic at two o'clock,three miles, reporting seven thousand feet."These advisories are issued on a "workloadpermitting" basis, which means that in somebusy situations the air traffic controller maycurtail the service. Automatic traffic advisoriesare generated by the Traffic Alert and CollisionAvoidance System, or TCAS (see ''TCAS: A Sys-

478

tern for Preventing Midair Collision" on page437 in this issue). These advisories are dis­played on a plan-position display located onthe aircraft instrument panel.

Traffic advisories improve visual acquisitionin two principal ways. First, they increase thefraction of the crew's time devoted to the visualsearch for traffic. Second, they concentrate vis­ual search in the proper direction. Aquantitativemeasure ofcrew performance with such adviso­ries was an objective of Lincoln Laboratory'swork in TCAS flight testing. Lincoln Laboratorybuilt and installed a prototype TCAS in a Cessna421 aircraft; the TCAS provided traffic adviso­ries through a special interface to a weatherradar CRT. The bearing accuracy of these advi­sories was approximately 7° (one standard de­viation) and the position of the traffic was up­dated once per second. Six different subjectpilots flew a variety of missions that resulted indata for 66 near-miss encounters.

The subjects were aware that, from time totime, a second test aircraft would conduct anintercept that would trigger the collision avoid­ance alarms. The interceptor was vectored byground radar to achieve a near-zero horizontalmiss distance with an altitude separation of200to 500 ft. From the viewpoint of the subjectpilots, the intercepts presented visual acquisi­tion problems similar to those ofactual collisionsituations. When the subjects used the TCAS,they visually acqUired in 57 of the 66 encoun­ters. The median range ofvisual acquisition was1.4 nmi. In five of the nine cases of acquisitionfailure, the subject aircraft received a climbadvisory from the TCAS and entered a nose-upattitude that prevented acquisition of the in­truder passing below.

To understand these results fully, we ana­lyzed the data in accordance with the visualacquisition model. A robust estimation tech­nique was developed to obtain the value of(3. Thetechnique was based on the fact that, accordingto the model, the probability of visual acquisi­tion within any time interval for which (3 isconstant can be determined from the timeintegral Q of the solid angle-contrast product ofthe target. Based on Eqs. 2 and 4, Q can be

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Andrews - Modeling ojAir-to-Air Visual Acquisition

,B =17,000/ster-s

• Unalerted Search

... Alerted Search (TeAS)

1.0

0.9

c 0.8g

0.7:i1i~

c- 0.6u«'0 0.5

~ 0.4:cell.D 0.30...a..

0.2

0.1

0.00

Even though the subjects had no traffic advi­sories to alert them to the time and direction ofapproach of the intercepts, additional care wastaken to prevent undue concentration on visualsearch. The tests were conducted as a generalstudy of single-pilot cockpit technique. Initialsubject briefings stated that the experimenterswere interested in seeing each pilot's individualtechnique, that there was no single standard forright and wrong, and that the subject should flyin as normal and relaxed a manner as possible.Data were collected in the cockpit by a safetypilot. Periodically this pilot would ask the sub­jects questions such as "Where is WorcesterAirport from our present location?" At severalpoints during the flight, the safety pilot wouldask subjects to rate their workload on a 1-to-9scale. They were also asked to call out all trafficwhen it was first seen; the times of sightingswere carefully recorded for later analysis. Thesubjects were told that other company trafficmight be operating in the area, but that theLincoln Laboratory control center would makesure that this traffic knew their location andkept clear of their altitude.

Fig. 3-Measured and modeled visual acquisition proba­bilities. The probability is defined in terms ofthe opportunityintegral-a measure of the angularsize of the visual targetand the duration of the search.

50 100 150 200Opportunity Integral (J1steradian-s)

t

Q(tl = f ~ exp ( -2.996 ~) dt. (7)

Since Qrepresents the opportunity for visualacquisition that the target has provided, Q iscalled the opportunity integral. The estimatedvalue of f3 is obtained by plotting observed acqui­sition probability as a function of Q and thenfinding the value of f3 that, when inserted in Eq.6, best fits the plotted data. Figure 3 shows theacquisition probabilities for TCAS flight testdata plotted as a function of Q. The value of f3that best fits this curve is 140,000/ster-s.

A numerical example places the value of f3 inperspective. Consider the acquisition rate for aBoeing 727 jet transport when seen head-on ata range of 3 nmi. The visual area of the aircraftis about 430 sq ft when seen from this angle. Ata range of 3 nmi, the aircraft will subtend anangle a of 1.3 x 10-6 ster. Thus the rate ofvisualacqUisition is A. =. f3a = 0.18/s, or approxi­mately 18% per each second of alerted search.Of course, the angular size of the aircraft isconstantly changing, and the integration indi­cated in Eq. 2 must be carried out in order tocompute the cumulative probability of visualacquisition over a period of time.

Unalerted-Search Flight Tests

defined as follows:

During the experiments with collision avoid­ance equipment, the focus of testing had beenon visual acquisition aided by automatic trafficadvisory avionics. Because most Visual FlightRule flights have no traffic advisory services, aneed remained to establish a baseline for unal­ertedsearch. With the supportofthe FAA, anewseries oftest flights were initiated to provide thisbaseline. In these tests, a group of 24 generalaviation pilots each flew a Beech Bonanza on atriangular cross-country flight of about 45minutes' duration. During this time, a Cessna421 aircraft conducted three intercepts; it flewboth over and under the test route to provide atarget for visual acquisition.

where

P(acg by t) = 1 - exp[-,BQ(tl] (6)

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Andrews - Modeling ojAir-to-Air Visual Acquisition

Time to Collision (s)

5 10 15 20 25 30 35 40 45 501.0

0.9

c 0.80:;::::;:§ 0.7::JC1"U~ 0.6co::J 0.5CIl

:>'0 0.4g

0.3:.aell.D 0.2ea..

0.1

0.00.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Range (nmi)

Fig. 4-Modelprediction ofvisual acquisitionprobabilityasa function of separation between aircraft for the Cerritosscenario. These curves assume unalerted visual search.

from the Piper. The acquisition probability in­creases significantly over the last minute offlight. At 12 s before collision, the probability ofvisual acquisition is 90%. Acquisition ofthe DC­9 is easier because it provides a visual targetthat is about eight times larger than the Piper.

This analysis shows that only in the lastminute prior to collision does a significantprobability for visual acquisition exist. Eventsthat occurred earlier (such as traffic advisoriesor cockpit distractions) are unlikely to have hadany effect on the failure to acquire.

A second significant resultwas to focus atten­tion on why the pilot of the Piper did not acquire.The results of the model apply only if normalvisual search is underway. Therefore, an un­usual factor must have existed that preventednormal search on board the Piper. In fact, theNTSB investigation revealed evidence that pilotKramer, who was navigating by visual groundreferences, had misidentified a freeway and hadstrayed from his intended course. At the time ofthe accident, he would probably have beenlooking out the left side of the aircraft forDisneyland, a prominent visual landmark usedby local pilots. If this idea was correct, it might

The test design appeared to be highly suc­cessful in preventing subjects from concentrat­ing unduly on visual search. Only two subjectsguessed that the Cessna 421 was the "companytraffic," and in neither case did this insight arisebefore the last intercept was over.

Data were obtained for 64 encounters. Visualacquisition was achieved in only 36 of theseencounters (56% of the total), and the medianacquisition range for the 36 encounters was0.99 nmi. Figure 3 provides a plot of the ac­quisition probability versus opportunity inte­gral for these unalerted-search data. The valueof f3 that best fits the data is approximately17,OOOjster-s.

In a comparison between the unalerted­search results and the alerted-search resultsfrom the earlier TCAS tests, the pilots who usedthe TCAS traffic advisories exhibited a f3 valuethat was higher by a factor of 8.2. The higherfactor implies that the presence of the TCAStraffic advisory increased search effectivenessby a factor of 8. In other words, one second ofsearch with the TCAS advisory was as effectiveas eight seconds of search with no alert.

Because neither aircraft in the Cerritos acci­dent had received a traffic advisory, the resultsof the unalerted-search flight tests were crucialto the analysis ofthat accident. Data provided bythe NTSB were used to determine the closingrate between the aircraft (271 knots) and thevisual areas of each aircraft as seen from theother (72 sq ft for the Piper and 588 sq ft for theDC-9). Figure 4 shows the results of the analy­sis. The bottom two curves describe the searchfor the Piper from the DC-9. The two-pilot curveis derived by assuming that the effect of the sec­ond pilot is to double the search performance forthe aircraft. The figure shows that the probabil­ity ofvisual acquisition is small until the last 15seconds or so before collision. The probability ofeffective collision avoidance (using 12-swarningtime as a criterion) is only 42%, even assumingthat both pilots are searching.

The top curve describes search for the DC-9

Analysis of the CerritosMidair Collision

480 The Lincoln Laboratory Journal. Volume 2. Number 3 (1989)

explain why the DC-9. approaching from 50° tothe right of the nose of the aircraft. would nothave been seen.

In the search for ways to prevent similaraccidents in the future, NTSB posed the follow­ing question: would the Cerritos accident havebeen prevented if the Aeromexico DC-9 had beenequipped with a TCAS? Because the Piper wasnot equipped with an altitude-reporting trans­ponder. the TCAS would have been unable toissue a resolution advisory. It would. however.have issued a traffic advisory that would haveaccurately depicted the angular position andrange of the Piper. A second set of calculationswere carried out under the assumption ofalerted search by the DC-9 crew. These calcula­tions showed that if they had been alerted by aTCAS traffic advisory. the crew would have hada 95% chance of seeing the Piper in time toavoid. The NTSB final report concluded that"had flight 498 been equipped with a TCAS.the accident might not have occurred." They re­commended that the FAA "expedite the de­velopment. operational evaluation. and finalcertification of the Traffic Alert and CollisionAvoidance System for installation and use incertificated air carrier aircraft (3)."

Conclusion

Maintenance of safe separation between air­craft is often dependent on the ability ofpilots tosee and avoid other traffic. But even whenweather conditions are favorable for visual sepa­ration, midair collisions often occur without thecrews ofeither colliding aircraft seeing the other.This fact raises central questions concerning theperformance that can be expected of air crews

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Andrews - Modeling ojAir-to-Air Visual Acquisition

and the potential benefits ofservices such as theautomatic traffic advisories available from colli­sion avoidance systems.

Although visual acquisition is a complexprocess, a useful statistical model of pilot per­formance has been developed at Lincoln Labora­tory. The model explains a variety ofexperimen­tal results that, without a common frameworkfor analysis, would be difficult to compare. Themodel is currently used by the FAA to evaluatecollision avoidance systems. airspace regula­tions, and waiver requests. The NTSB has alsoused it to provide insight into crew performancein accident scenarios. The model has a signifi­cant limitation: it can be applied only where thepilot performance level can be assumed to ap­proximate that for which flight test data areavailable. Extension of the model to other situ­ations (such as high-workload landing phase ormilitary training situations) may require pilotperformance testing with either actual flight orrealistic flight simulators.

References

1. J.W. Andrews. "Air-to-Air Visual Acquisition Perfor­mance with Pilot Warning Instruments (PWl)," ProjectReport ATC-73. Lincoln Laboratory (25 Apr. 1977). FAA­RD-77-30.

2. W.O. Howell. "Determination ofDaytime Conspicuity ofTransport Aircraft." Technical Development Report No.304. Civil Aeronautics Administration (May 1957).

3. A. MiIIhollon. J. Lyons. and W. Graham. "Air-to-AirVisual Detection Data." Interim Report. Control DataCorporation (Apr. 1973). FAA-RD-73-40.

4. J.W. Andrews. "Air-to-Air Visual Acquisition Perfor­mance with TCAS II." Project Report ATC-l30, LincolnLaboratory (27 July 1984), DOT/FAA/PM-84-17.

5. "Collision of Aeronaves de Mexico. S.A. McDonnellDouglas DC-9-32. XA-JED and Piper PA-28-181.N4891 F, Cerritos. California. August 31, 1986."AircraJtAccident Report No. NTSB/AAR-8l /07. National Trans­portation Safety Board (7 July 1987).

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Andrews - Modeling ojAir-to-Air Visual Acquisition

JOHN ANDREWS is an as­sistant group leader in theSystem Design and Evalu­ation Group and is currentlyresponsible for technicalmanagement of the Termi­nal Air Traffic Control Auto­

mation (TATCA) Program. For 14 years. he was involved inthe development ofcollision avoidance systems. His workonthe Traffic Alert and Collision Avoidance System (TeAS)resulted in contributions to aircraft-altitude tracking tech­niques. analysis of computer logiC. human-subject flighttesting. and the analysis ofpilotvisual acquisition perform­ance. John has served as a consultant to the NationalTransportation Safety Board in the investigation of midaircollisions. including the Cerritos accident. He has a B.S.degree in physics from the Georgia Institute ofTechnology.and an S.M. degree in aeronautical engineering from MIT.

482 The Lincoln Laboratory Journal. Volume 2. Number 3 (1989)

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