J.W. Andrews Modeling ofAir-to-Air Visual Acquisition A mathematical model ofair-to-airvisual acquisition has been developed and validated ina series offlight testsatLincoln Laboratory. Themodel describes thevisual acquisition process as a nonhomogeneous Poisson process in which the probability of visual acquisition per unit of time is proportional to the solid angle subtended by the target. The model has proven useful in the investigation ofactual midair collisions, as well as in the evaluation of collision avoidance systems. The Cerritos Midair Collision takingly transcribed at the NTSB laboratory in Washington, D.C. Air traffic personnel, airline August 31, 1986 was a typical warm, sunny officials, eyewitnesses, friends, and relatives of summer dayin southern California. At 11 a.m., the pilots were interviewed. Among the many three people boarded a single-engine Piper important questions to answer was a familiar Archer airplane at the Torrance, Calif., airfield one to the NTSB investigators: why didn't the and departed on a flight to the resortarea ofBig two aircraft see each other in time to avoid a Bear, Calif. The pilot of the Piper was William collision? Kramer, a 53-year-old metallurgist who owned In pursuit ofthe visual acquisition question, the aircraft. Onboardwere his daughterandhis NTSB investigator Malcolm Brenner contacted wife. The Kramers planned a day ofvacation at Lincoln Laboratory about studies conducted as Big Bear and a return flight to Torrance later part ofLincoln Laboratory's collision avoidance thatevening. As the Piperwas climbingoutfrom work. Lincoln Laboratory had developed a Torrance, Aeromexico flight 498, a DC-9 air­ mathematical model of visual acquisition and craft, was descending into Los Angeles on the hadjustcompleteda seriesofflight tests directly last leg of a regularly scheduled flight that had applicableto Cerritos. Lincolnagreed to support originated in Mexico City. In the radar control the NTSB, and the details of the Cerritos acci­ facility at Los Angeles International Airport, air dent were soon being analyzed with the Lincoln traffic controllerWalterWhite cleared Aeromex­ model. ico to descend for the approach. No one seemed to suspect that anything was amiss until, at Origin of the Model 6,500 feet, the Piper collided with the tail ofthe DC-9 (Fig. 1). The Kramers were fatally injured Visual acquisition is not a deterministic pro­ inthe collision; thePiperspunoutofcontroland cess; iUs described bymodels thatareprobabil­ slammed into a school yard in the suburban istic in nature. For small targets, acqUisition town of Cerritos. The DC-9 plunged to earth in takes place onlywhen the direction ofthe pilot's a residentialarea, killingall 64personsonboard visual search is aligned almost exactly with the together with 15 people on the ground (Fig. 2). line ofsightto the target. Therandomness ofthe In the months that followed, investigators searchprocess, combined with the randomness from the National Transportation Safety Board of the detection probability at a given angular (NTSB) carefully reconstructed the events that offset, causes acquisition times to vary greatly, led to the tragedy. The wreckage was reas­ even under identical visual conditions. An air­ sembled, measured, and photographed. The craft on a collision course is usually well above cockpitvoice recorderwas recovered andpains- the visual resolution threshold ofthe eye before The Lincoln Laboratory Journal Volume 2, Number 3 (1989) 475 Andrews - Modeling ojAir-to-Air Visual Acquisition Fig. 1-lmpactgeometry for the Cerritos, Calif., midair collision between an Aeromexico DC-9 anda PiperArcher, on August31, 1986. According to eyewitnesses, neitheraircraft took anyevasive action before impact. it is acquired. Pilots have described the phe­ The immediate question to be answered was nomenon ofacquisitionwell above the threshold the degree of benefit from the automatic traffic by saying that targets seem to "pop up outofthe advisories provided by the collision avoidance sky." In reality, the targets are visible for some equipment. At first glance, it might seem that time before the search results in acquisition. the benefits could be characterized by some The Lincoln Laboratory model originated simple statistic such as the range at which the dUring testingofa ground-based collision avoid­ approaching traffic was seen. But two difficul­ ance system known as the Automatic Traffic ties arose. The first problemwas that the visual Advisory and Collision Avoidance System. Dur­ acquisition process was highly variable and ing the flight tests, subject pilots flew near­ thus required a suitable statistical approach. collision encounters in two Lincoln Laboratory The second and more significant problem was aircraftbased atL.G. Hanscom Field in Bedford, that manyfactors affected the difficulty ofvisual Mass. Radar measured the paths ofthe aircraft, acquisition. Among these factors were the speed and radio transmission determined the time and geometry ofapproach, the size ofthe target when the pilots visually acquired each other. By aircraft, the timing and accuracy of traffic ad­ correlating the speeds and distances with the visories, and the preVailing atmospheric visibil­ times ofvisual acquisition, the flight tests pro­ ity (visual range). How could visual acqUisition duced quantitative measures of pilot visual data collected at one set of speeds against a acquisition perfonnance. single aircraft be applied to the general problem 476 The Lincoln Laboratory Journal. Volume 2. Number 3 (1989) Andrews - Modeling ojAir-to-Air Visual Acquisition Unfortunately, the air-to-air visual acquisition process is complicated because Ais not constant with time. In fact. A increases steadily as two aircraft on a collision course approach closer together. The cumulative probability of visual acqUisition must then be written P(acq by t,) = 1- exp[ -IA(tldtl (2) This equation describes a nonhomogeneous Poisson process and provides a general mathe­ maticalformulation thatcanbe used to describe Fig. 2-Aftermath of the Cerritos, Calif., midair collision. many different visual search situations. Fifteen people on the ground were killed in addition to 64 To apply Eq. 2. an expression for Amust be people on the airplane. The accident occurred on a clear day when the reported visual range was 14 nmi. found. Several setsoftestdatawere examinedin addition to the Lincoln Laboratory data. The other sets of data included a classic flight test conducted in DC-3s by Wayne Howell for the involving a range of speeds and a variety of Civil Aeronautics Administration in 1957 [2], aircraft? If the conclusions were to be generally and data collected by Control Data Corporation applicable. it would be necessary to devise a dUring a series of air-to-air photographic mis­ means of extrapolating experimental results to sions in 1973 [3). When properly analyzed, all other conditions. The mathematical model of data sets supported the conclusion that the visual acquisition was developed to serve this visual acquisition rate is proportional to the purpose. angular size of the visual target. That is, The first step in developing the model was to establish a mathematical formalism that could (3) serve as a powerful yet general basis for analy­ sis. The key to such a formalism proved to be the where A is the visual area presented by the use oftheinstantaneousvisualacquisition rate. target aircraft, ris the range to the target. and f3 At any given instant. the visual acquisition rate is a constant to be determined. Acan be defined as follows: The above expression is suitable only when theatmosphere is sufficientlyclearatthe ranges ?c(t) = lim P(acq in M) . (1) ofinterest. In fog or haze. the contrast oftargets M---70 tlt can decrease rapidly with range; this depend­ This rate is the probability ofvisual acquisi­ ency must be included iff3 is to be modeled as a tion perinstant time. IfA = O.02/s. for example, constant. Laboratoryexperiments showthat the then the probability ofvisual acquisition in one detectability of a static target is actually deter­ second of search is approximately 2%. The mined bytheproductofthe target size and target cumulative probability of visual acquisition is contrast. This recognition suggests that the obtained by evaluating the acquisition proba­ size-contrast product should be substituted for bilities for each instant as the target aircraft the size expression in Eq. 3. A relationship approaches. If Awere constant. then the result known as Koschmieder's law states that. in a would be a set of simple equations that corre­ homogeneous atmosphere, apparent contrast spond to the classic Poisson process. a process degrades exponentiallywithvisual range. Thus. used widely to describe physical processes such if Co is the inherent contrast of the target with as the rate of fission of radioactive isotopes. its background, then at range r the apparent The Lincoln Laboratory Journal. VoLume 2. Number 3 (1989) 477 Andrews - Modeling ojAir-to-Air Visual Acquisition contrast is reduced to tern for Preventing Midair Collision" on page ~). 437 in this issue). These advisories are dis­ Co exp(-2.996 played on a plan-position display located on the aircraft instrument panel. The constant -2.996 arises from the defini­ Traffic advisories improve visual acquisition tion of visual range as the range at which in two principal ways. First, they increase the apparent contrast is reduced to 5% ofthe inher­ fraction of the crew's time devoted to the visual ent contrast. search