PART I. FLIGHT DENSITIES AND THEIR DETERMINATION

A. LUNAR OBSERVATIONS OF BIRDS AND THE FLIGHT DENSITY CONCEPT

The subject matter of this paper is wholly ornithological. It is written for the zoologist interested in the activities of birds. But its bases, the principles that make it possible, lie in other fields, including such rather advanced branches of mathematics as analytical geometry, spherical geometry, and differential calculus. No exhaustive exposition of the problem is practicable, that does not take for granted some previous knowledge of these disciplines on the part of all readers.

There are, however, several levels of understanding. It is possible to appreciate _what_ is being done without knowing _how_ to do it; and it is possible to learn how to carry out the successive steps of a procedure without entirely comprehending _why_. Some familiarity with the concepts underlying the method is essential to a full understanding of the results achieved, and details of procedure must be made generally available if the full possibilities of the telescopic approach are to be realized. Without going into proof of underlying propositions or actual derivation of formulae, I shall accordingly present a discussion of the general nature of the problem, conveyed as much as possible in terms of physical visualization. The development begins with the impressions of the student when he first attempts to investigate the movements of birds by means of the moon.

_What the Observer Sees_

Watched through a 20-power telescope on a cloudless night, the full moon shines like a giant plaster hemisphere caught in the full glare of a floodlight. Inequalities of surface, the rims of its craters, the tips of its peaks, gleam with an almost incandescent whiteness; and even the darker areas, the so-called lunar seas, pale to a clear, glowing gray.

Against this brilliant background, most birds passing in focus appear as coal-black miniatures, only 1/10 to 1/30 the apparent diameter of the moon. Small as these silhouettes are, details of form are often beautifully defined--the proportions of the body, the shape of the tail, the beat of the wings. Even when the images are so far away that they are pin-pointed as mere flecks of black against the illuminated area, the normal eye can follow their progress easily. In most cases the birds are invisible until the moment they "enter," or pass opposite, the rim of the moon and vanish the instant they reach the other side. The interval between is likely to be inestimably brief. Some birds seem fairly to flash by; others, to drift; yet seldom can their passing be counted in seconds, or even in measureable fractions of seconds. During these short glimpses, the flight paths tend to lie along straight lines, though occasionally a bird may be seen to undulate or even to veer off course.

Now and again, in contrast to this typical picture, more eerie effects may be noted. Some of them are quite startling--a minute, inanimate-looking object drifting passively by like a corpuscle seen in the field of a microscope; a gigantic wing brushing across half the moon; a ghost-like suggestion of a bird so transparent it seems scarcely more than a product of the imagination; a bird that pauses in mid-flight to hang suspended in the sky; another that beats its way ineffectually forward while it moves steadily to the side; and flight paths that sweep across the vision in astonishingly geometric curves. All of these things have an explanation. The "corpuscle" is possibly a physical entity of some sort floating in the fluid of the observer's eye and projected into visibility against the whiteness of the moon. The winged transparency may be an insect unconsciously picked up by the unemployed eye and transferred by the _camera lucida_ principle to the field of the telescope. It may be a bird flying very close, so drastically out of focus that the observer sees right through it, as he would through a pencil held against his nose. The same cause, operating less effectively, gives a characteristic gray appearance with hazy edges to silhouettes passing just beneath the limits of sharp focus. Focal distortions doubtless also account for the precise curvature of some flight paths, for this peculiarity is seldom associated with distinct images. Suspended flight and contradictory directions of drift may sometimes be attributable to head winds or cross winds but more often are simply illusions growing out of a two-dimensional impression of a three-dimensional reality.

Somewhat more commonplace are the changes that accompany clouds. The moon can be seen through a light haze and at times remains so clearly visible that the overcast appears to be behind, instead of in front of, it. Under these circumstances, birds can still be readily discerned. Light reflected from the clouds may cause the silhouettes to fade somewhat, but they retain sufficient definition to distinguish them from out-of-focus images. On occasion, when white cloud banks lie at a favorable level, they themselves provide a backdrop against which birds can be followed all the way across the field of the telescope, whether or not they directly traverse the main area of illumination.

_Types of Data Obtained_

The nature of the observations just described imposes certain limitations on the studies that can be made by means of the moon. The speed of the birds, for instance, is utterly beyond computation in any manner yet devised. Not only is the interval of visibility extremely short, but the rapidity with which the birds go by depends less on their real rate of motion than on their proximity to the observer. The identification of species taking part in the migration might appear to offer more promise, especially since some of the early students of the problem frequently attempted it, but there are so many deceptive elements to contend with that the results cannot be relied upon in any significant number of cases. Shorn of their bills by the diminution of image, foreshortened into unfamiliar shape by varying angles of perspective, and glimpsed for an instant only, large species at distant heights may closely resemble small species a few hundred feet away. A sandpiper may appear as large as a duck; or a hawk, as small as a sparrow. A goatsucker may be confused with a swallow, and a swallow may pass as a tern. Bats, however, can be consistently recognized, if clearly seen, by their tailless appearance and the forward tilt of their wings, as well as by their erratic flight. And separations of nocturnal migrants into broad categories, such as seabirds and passerine birds, are often both useful and feasible.

It would be a wonderful convenience to be able to clock the speed of night-flying birds accurately and to classify them specifically, but neither of these things is indispensable to the general study of nocturnal migration, nor as important as the three kinds of basic data that _are_ provided by telescopes directed at the moon. These concern:--(1) the direction in which the birds are traveling; (2) their altitude above the earth; (3) the number per unit of space passing the observation station.

Unfortunately none of these things can be perceived directly, except in a very haphazard manner. Direction is seen by the observer in terms of the slant of a bird's pathway across the face of the moon, and may be so recorded. But the meaning of every such slant in terms of its corresponding compass direction on the plane of the earth constantly changes with the position of the moon. Altitude is only vaguely revealed through a single telescope by the size and definition of images whose identity and consequent real dimensions are subject to serious misinterpretation, for reasons already explained. The number of birds per unit of space, seemingly the easiest of all the features of migration to ascertain, is actually the most difficult, requiring a prior knowledge of both direction and altitude. To understand why this is so, it will be necessary to consider carefully the true nature of the field of observation.

_The Changing Field of Observation_

Most of the observations used in this study were made in the week centering on the time of the full moon. During this period the lunar disc progresses from nearly round to round and back again with little change in essential aspect or apparent size. To the man behind the telescope, the passage of birds looks like a performance in two dimensions taking place in this area of seemingly constant diameter--not unlike the movement of insects scooting over a circle of paper on the ground. Actually, as an instant's reflection serves to show, the two situations are not at all the same. The insects are all moving in one plane. The birds only appear to do so. They may be flying at elevations of 500, 1000, or 2000 feet; and, though they give the illusion of crossing the same illuminated area, the actual breadth of the visible space is much greater at the higher, than at the lower, level. For this reason, other things being equal, birds nearby cross the moon much more swiftly than distant ones. The field of observation is not an area in the sky but a volume in space, bounded by the diverging field lines of the observer's vision. Specifically, it is an inverted cone with its base at the moon and its vertex at the telescope.

Since the distance from the moon to the earth does not vary a great deal, the full dimensions of the Great Cone determined by the diameter of the moon and a point on the earth remain at all times fairly constant. Just what they are does not concern us here, except as regards the angle of the apex (roughly 1/2°), because obviously the effective field of observation is limited to that portion of the Great Cone below the maximum ceiling at which birds fly, a much smaller cone, which I shall refer to as the Cone of Observation (Figure 1).

The problem of expressing the number of passing birds in terms of a definite quantity of space is fundamentally one of finding out the critical dimensions of this smaller cone. The diameter at any distance from the observer may be determined with enough accuracy for our purposes simply by multiplying the distance by .009, a convenient approximation of the diameter of the moon, expressed in radians (see Figure 2). One hundred feet away, it is approximately 11 inches; 1000 feet away, nine feet; at one mile, 48 feet; at two miles, 95 feet. Estimating the effective length of the field of observation presents more formidable difficulties, aggravated by the fact that the lunar base of the Great Cone does not remain stationary. The moon rises in the general direction of east and sets somewhere in the west, the exact points where it appears and disappears on the horizon varying somewhat throughout the year. As it drifts across the sky it carries the cone of observation with it like the slim beam of an immense searchlight slowly probing space. This situation is ideal for the purpose of obtaining a random sample of the number of birds flying out in the darkness, yet it involves great complications; for the size of the sample is never at two consecutive instants the same. The nearer the ever-moving great cone of the moon moves toward a vertical position, the nearer its intersection with the flight ceiling approaches the observer, shortening, therefore, the cone of observation (Figure 3). The effect on the number of birds seen is profound. In extreme instances it may completely reverse the meaning of counts. Under the conditions visualized in Figure 3, the field of observation at midnight is only one-fourth as large as the field of observation earlier in the evening. Thus the twenty-four birds seen from 7 to 8 P. M., represent not twice as many birds actually flying per unit of space as the twelve observed from 11:30 to 12:30 A. M., but only half the amount. Figure 4, based on observations at Ottumwa, Iowa, on the night of May 22-23, shows a similar effect graphically. Curve A represents the actual numbers of birds per hour seen; Curve B shows the same figures expressed as flight densities, that is, corrected to take into account the changing size of the field of observation. It will be noted that the trends are almost exactly opposite. While A descends, B rises, and _vice-versa_. In this case, inferences drawn from the unprocessed data lead to a complete misinterpretation of the real situation.

Nor does the moon suit our convenience by behaving night after night in the same way. On one date we may find it high in the sky between 9 and 10 P. M.; on another date, during the same interval of time, it may be near the horizon. Consequently, the size of the cone is different in each case, and the direct comparison of flights in the same hour on different dates is no more dependable than the misleading comparisons discussed in the preceding paragraph.

The changes in the size of the cone have been illustrated in Figure 3 as though the moon were traveling in a plane vertical to the earth's surface, as though it reached a point directly over the observer's head. In practice this least complicated condition seldom obtains in the regions concerned in this study. In most of the northern hemisphere, the path of the moon lies south of the observer so that the cone is tilted away from the vertical plane erected on the parallel of latitude where the observer is standing. In other words it never reaches the zenith, a point directly overhead. The farther north we go, the lower the moon drops toward the horizon and the more, therefore, the cone of observation leans away from us. Hence, at the same moment, stationed on the same meridian, two observers, one in the north and one in the south, will be looking into different effective volumes of space (Figure 5).

As a further result of its inclination, the cone of observation, seldom affords an equal opportunity of recording birds that are flying in two different directions. This may be most easily understood by considering what happens on a single flight level. The plane parallel to the earth representing any such flight level intersects the slanting cone, not in a circle, but in an ellipse. The proportions of this ellipse are very variable. When the moon is high, the intersection on the plane is nearly circular; when the moon is low, the ellipse becomes greatly elongated. Often the long axis may be more than twice the length of the short axis. It follows that, if the long axis happens to lie athwart the northward direction of flight and the short axis across the eastward direction, we will get on the average over twice as large a sample of birds flying toward the north as of birds flying toward the east.

In summary, whether we wish to compare different stations, different hours of the night, or different directions during the same hour of the night, no conclusions regarding even the relative numbers of birds migrating are warranted, unless they take into account the ever-varying dimensions of the field of observation. Otherwise we are attempting to measure migration with a unit that is constantly expanding or contracting. Otherwise we may expect the same kind of meaningless results that we might obtain by combining measurements in millimeters with measurements in inches. Some method must be found by which we can reduce all data to a standard basis for comparison.

_The Directional Element in Sampling_

In seeking this end, we must immediately reject the simple logic of sampling that may be applied to density studies of animals on land. We must not assume that, since the field of observation is a volume in space, the number of birds therein can be directly expressed in terms of some standard volume--a cubic mile, let us say. Four birds counted in a cone of observation computed as 1/500 of a cubic mile are not the equivalent of 500 × 4, or 2000, birds per cubic mile. Nor do four birds flying over a sample 1/100 of a square mile mathematically represent 400 birds passing over the square mile. The reason is that we are not dealing with static bodies fixed in space but with moving objects, and the objects that pass through a cubic mile are not the sum of the objects moving through each of its 500 parts. If this fact is not immediately apparent, consider the circumstances in Figures 6 and 7, illustrating the principle as it applies to areas. The relative capacity of the sample and the whole to intercept bodies in motion is more closely expressed by the ratio of their perimeters in the case of areas and the ratio of their surface areas in the case of volumes. But even these ratios lead to inaccurate results unless the objects are moving in all directions equally (see Figure 8). Since bird migration exhibits strong directional tendencies, I have come to the conclusion that no sampling procedure that can be applied to it is sufficiently reliable short of handling each directional trend separately.

For this reason, the success of the whole quantitative study of migration depends upon our ability to make directional analyses of primary data. As I have already pointed out, the flight directions of birds may be recorded with convenience and a fair degree of objectivity by noting the slant of their apparent pathways across the disc of the moon. But these apparent pathways are seldom the real pathways. Usually they involve the transfer of the flight line from a horizontal plane of flight to a tilted plane represented by the face of the moon, and so take on the nature of a projection. They are clues to directions, but they are not the directions themselves. For each compass direction of birds flying horizontally above the earth, there is one, and only one, slant of the pathway across the moon at a given time. It is possible, therefore, knowing the path of a bird in relation to the lunar disc and the time of the observation, to compute the direction of its path in relation to the earth. The formula employed is not a complicated one, but, since the meaning of the lunar coördinates in terms of their corresponding flight paths parallel to the earth is constantly changing with the position of the moon, the calculation of each bird's flight separately would require a tremendous amount of time and effort.

Whatever we do, computed individual flight directions must be frankly recognized as approximations. Their anticipated inaccuracies are not the result of defects in the mathematical procedure employed. This is rigorous. The difficulty lies in the impossibility of reading the slants of the pathways on the moon precisely and in the three-dimensional nature of movement through space. The observed coördinates of birds' pathways across the moon are the projected product of two component angles--the compass direction of the flight and its slope off the horizontal, or gradient. These two factors cannot be dissociated by any technique yet developed. All we can do is to compute what a bird's course would be, if it were flying horizontal to the earth during the interval it passes before the moon. We cannot reasonably assume, of course, that all nocturnal migration takes place on level planes, even though the local distractions so often associated with sloping flight during the day are minimized in the case of migrating birds proceeding toward a distant destination in darkness. We may more safely suppose, however, that deviations from the horizontal are random in nature, that it is mainly a matter of chance whether the observer happens to see an ascending segment of flight or a descending one. Over a series of observations, we may expect a fairly even distribution of ups and downs. It follows that, although departures from the horizontal may distort individual directions, they tend to average out in the computed trend of the mean. The working of this principle applied to the undulating flight of the Goldfinch (_Spinus_) is illustrated in Figure 9.

Since _individually_ computed directions are not very reliable in any event, little is to be lost by treating the observed pathways in groups. Consequently, the courses of all the birds seen in a one-hour period may be computed according to the position of the moon at the middle of the interval and expressed in terms of their general positions on the compass, rather than their exact headings. For this latter purpose, the compass has been divided into twelve fixed sectors, 22-1/2 degrees wide. The trends of the flight paths are identified by the mid-direction of the sector into which they fall. The sectoring method is described in detail in the section on procedures.

The problem remains of converting the number of birds involved in each directional trend to a fixed standard of measurement. Figure 7A contains the partial elements of a solution. All of the west-east flight paths that cross the large square also cross one of its mile-long sides and suggest the practicability of expressing the amount of migration in any certain direction in terms of the assumed quantity passing over a one-mile line in a given interval of time. However, many lines of that length can be included within the same set of flight paths (Figure 10); and the number of birds intercepted depends in part upon the orientation of the line. The 90° line is the only one that fully measures the amount of flight per linear unit of front; and so I have chosen as a standard an imaginary mile on the earth's surface lying at right angles to the direction in which the birds are traveling.

_Definitions of Flight Density_

When the count of birds in the cone of observation is used as a sample to determine the theoretical number in a sector passing over such a mile line, the resulting quantity represents what I shall call a Sector Density. It is one of several expressions of the more general concept of Flight Density, which may be defined as the passage of migration past an observation station stated in terms of the theoretical number of birds flying over a one-mile line on the earth's surface in a given interval of time. Note that a flight density is primarily a theoretical number, a statistical expression, a _rate_ of passage. It states merely that birds were moving through the effective field of observation at the _rate_ of so many per mile per unit of time. It may or may not closely express the amount of migration occurring over an actual mile or series of miles. The extent to which it does so is to be decided by other general criteria and by the circumstances surrounding a given instance. Its basic function is to take counts of birds made at different times and at different places, in fields of observation of different sizes, and to put them on the statistically equal footing that is the first requisite of any sound comparison.

The idea of a one-mile line as a standard spacial measurement is an integral part of the basic concept, as herein propounded. But, within these limitations, flight density may be expressed in many different ways, distinguished chiefly by the directions included and the orientation of the one-mile line with respect to them. Three such kinds of density have been found extremely useful in subsequent analyses and are extensively employed in this paper: Sector, Net Trend, and Station Density, or Station Magnitude.

Sector Density has already been referred to. It may be defined as the flight density within a 22-1/2° directional spread, or sector, measured across a one-mile line lying at right angles to the mid-direction of the sector. It is the basic type of density from the point of view of the computer, the others being derived from it. In analysis it provides a means of comparing directional trends at the same station and of studying variation in directional fanning.

Net Trend Density represents the maximum net flow of migration over a one-mile line. It is found by plotting the sector densities directionally as lines of thrust, proportioned according to the density in each sector, and using vector analysis to obtain a vector resultant, representing the density and direction of the net trend. The mile line defining the spacial limits lies at right angles to this vector resultant, but the density figure includes all of the birds crossing the line, not just those that do so at a specified angle. Much of the directional spread exhibited by sector densities undoubtedly has no basis in reality but results from inaccuracies in coördinate readings and from practical difficulties inherent in the method of computation. By reducing all directions to one major trend, net trend density has the advantage of balancing errors one against the other and may often give the truer index to the way in which the birds are actually going. On the other hand, if the basic directions are too widely spread or if the major sector vectors are widely separated with little or no representation between, the net trend density may become an abstraction, expressing the idea of a mean direction but pointing down an avenue along which no migrants are traveling. In such instances, little of importance can be learned from it. In others, it gives an idea of general trends indispensable in comparing station with station to test the existence of flyways and in mapping the continental distribution of flight on a given night to study the influence of weather factors.

Station Density, or Station Magnitude, represents all of the migration activity in an hour in the vicinity of the observation point, regardless of direction. It expresses the sum of all sector densities. It includes, therefore, the birds flying at right angles over several one-mile lines. One way of picturing its physical meaning is to imagine a circle one-mile in diameter lying on the earth with the observation point in the center. Then all of the birds that fly over this circle in an hour's time constitute the hourly station density. While its visualization thus suggests the idea of an area, it is derived from linear expressions of density; and, while it involves no limitation with respect to direction, it could not be computed without taking every component direction into consideration. Station density is adapted to studies involving the total migration activity at various stations. So far it has been the most profitable of all the density concepts, throwing important light on nocturnal rhythm, seasonal increases in migration, and the vexing problem of the distribution of migrating birds in the region of the Gulf of Mexico.

Details of procedure in arriving at these three types of flight density will be explained in Section B of this discussion. For the moment, it will suffice to review and amplify somewhat the general idea involved.

_Altitude as a Factor in Flight Density_

A flight density, as we have seen, may be defined as the number of birds passing over a line one mile long; and it may be calculated from the number of birds crossing the segment of that line included in an elliptical cross-section of the cone of observation. It may be thought of with equal correctness, without in any way contradicting the accuracy of the original definition, as the number of birds passing through a vertical plane one mile long whose upper limits are its intersection with the flight ceiling and whose base coincides with the one mile line of the previous visualization. From the second point of view, the sample becomes an area bounded by the triangular projection of the cone of observation on the density plane. The dimensions of two triangles thus determined from any two cones of observation stand in the same ratio as the dimensions of their elliptical sections on any one plane; so both approaches lead ultimately to the same result. The advantage of this alternative way of looking at things is that it enables us to consider the vertical aspects of migration--to comprehend the relation of altitude to bird density.

If the field of observation were cylindrical in shape, if it had parallel sides, if its projection were a rectangle or a parallelogram, the height at which birds are flying would not be a factor in finding out their number. Then the sample would be of equal breadth throughout, with an equally wide representation of the flight at all levels. Since the field of observation is actually an inverted cone, triangular in section, with diverging sides, the opportunity to detect birds increases with their distance from the observer. The chances of seeing the birds passing below an elevation midway to the flight ceiling are only one-third as great as of seeing those passing above that elevation, simply because the area of that part of the triangle below the mid-elevation is only one-third as great as the area of that part above the mid-elevation. If we assume that the ratio of the visible number of birds to the number passing through the density plane is the same as the ratio of the triangular section of the cone to the total area of the plane, we are in effect assuming that the density plane is made up of a series of triangles the size of the sample, each intercepting approximately the same number of birds. We are assuming that the same number of birds pass through the inverted triangular sample as through the erect and uninvestigable triangle beside it (as in Figure 11, Diagram II). In reality, the assumption is sound only if the altitudinal distribution of migrants is uniform.

The definite data on this subject are meagre. Nearly half a century ago, Stebbins worked out a way of measuring the altitude of migrating birds by the principle of parallax. In this method, the distance of a bird from the observers is calculated from its apparent displacement on the moon as seen through two telescopes. Stebbins and his colleague, Carpenter, published the results of two nights of observation at Urbana, Illinois (Stebbins, 1906; Carpenter, 1906); and then the idea was dropped until 1945, when Rense and I briefly applied an adaptation of it to migration studies at Baton Rouge. Results have been inconclusive. This is partly because sufficient work has not been done, partly because of limitations in the method itself. If the two telescopes are widely spaced, few birds are seen by both observers, and hence few parallaxes are obtained. If the instruments are brought close together, the displacement of the images is so reduced that extremely fine readings of their positions are required, and the margin of error is greatly increased. Neither alternative can provide an accurate representative sample of the altitudinal distribution of migrants at a station on a single night. New approaches currently under consideration have not yet been perfected.

Meanwhile the idea of uniform vertical distribution of migrants must be dismissed from serious consideration on logical grounds. We know that bird flight cannot extend endlessly upward into the sky, and the notion that there might be a point to which bird density extends in considerable magnitude and then abruptly drops off to nothing is absurd. It is far more likely that the migrants gradually dwindle in number through the upper limits at which they fly, and the parallax observations we have seem to support this view.

Under these conditions, there would be a lighter incidence of birds in the sample triangle than in the upright triangle beside it (Figure 11, Diagram III). Compensation can be made by deliberately scaling down the computed size of the sample area below its actual size. A procedure for doing this is explained in Figure 11. If it were applied to present altitudinal data, it would place the computational flight ceiling somewhere below 4000 feet. In arriving at the flight densities used in this paper, however, I have used an assumed ceiling of one mile. When the altitude factor is thus assigned a value of 1, it disappears from the formula, simplifying computations. Until the true situation with respect to the vertical distribution of flight is better understood, it seems hardly worthwhile to sacrifice the convenience of this approximation to a rigorous interpretation of scanty data. This particular uncertainty, however, does not necessarily impair the analytical value of the computations. Provided that the vertical pattern of migration is more or less constant, flight densities still afford a sound basis for comparisons, wherever we assume the upper flight limits to be. Raising or lowering the flight ceiling merely increases or reduces all sample cones or triangles proportionately.

A more serious possibility is that the altitudinal pattern may vary according to time or place. This might upset comparisons. If the divergencies were severe enough and frequent enough, they could throw the study of flight densities into utter confusion.

This consideration of possible variation in the altitudinal pattern combines with accidents of sampling and the concessions to perfect accuracy, explained on pages 379-385, to give to small quantities of data an equivocal quality. As large-scale as the present survey is from one point of view, it is only a beginning. Years of intensive work and development leading to a vast accumulation of data must elapse before the preliminary indications yet discernible assume the status of proved principles. As a result, much of the discussion in