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 deg.), 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).
[Illustration: FIG. 1. The field of observation, showing
its two-dimensional aspect as it appears to the observer and
its three-dimensional actuality. The breadth of the cone is
greatly exaggerated.]
[Illustration: FIG. 2. Method for determining the diameter
of the cone at any point. The angular diameter of the moon
may be expressed in radians, or, in other words, in terms of
lengths of arc equivalent to the radius of a circle. In the
diagram, the arc between C and E, being equivalent to the
radius CO, represents a radian. If we allow the arc between A
and B to be the diameter of the moon, it is by astronomical
calculation about .009 radian, or .009 CO. This ratio will
hold for any smaller circle inscribed about the center O;
that is, the arc between A'B' equals .009 C'O. Thus the width
of the cone of observation at any point, expressed in degrees
of arc, is .009 of the axis of the cone up to that point. The
cone is so slender that the arc between A and B is
essentially equal to the chord AB. Exactly the same
consideration holds true for the smaller circle where the
chord A'B' represents part of the flight ceiling.]
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.
[Illustration: FIG. 3. Temporal change in the effective
size of the field of observation. The sample sections, A and
B, represent the theoretical densities of flight at 8:20 and
12:00 P. M., respectively. Though twice as many birds are
assumed to be in the air at midnight when the moon is on its
zenith (Z) as there were at the earlier hour, only half as
many are visible because of the decrease in size of the cone
of observation.]
[Illustration: FIG. 4. Migration at Ottumwa, Iowa, on the
night of May 22-23, 1948. Curve A is a graphic representation
of the actual numbers of birds seen hourly through the
telescope. Curve B represents the same figures corrected for
the variation in the size of the cone of observation. The
dissimilarity in the two curves illustrates the deceptive
nature of untreated telescopic counts.]
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).
[Illustration: FIG. 5. Geographical variation in the size
of the cone of observation. The cones A and B represent the
effective fields of observation at two stations situated over
1,200 miles apart. The portions of the great cones included
here appear nearly parallel, but if extended far enough would
be found to have a common base on the moon. Because of the
continental scale of the drawing, the flight ceiling appears
as a curved surface, equidistant above each station. The
lines to the zenith appear to diverge, but they are both
perpendicular to the earth. Although the cones are shown at
the same instant in time, and have their origin on the same
meridian, the dimensions of B are less than one-half as great
as those of A, thus materially decreasing the opportunity to
see birds at the former station. This effect results from the
different slants at which the zenith distances cause the
cones to intersect the flight ceiling. The diagram
illustrates the principle that northern stations, on the
average, have a better chance to see birds passing in their
vicinity than do southern stations.]
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 x 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.
[Illustration: FIG. 6. The problem of sampling migrating
birds. The large square in the diagram may be thought of as a
square mile on the earth's surface, divided into four equal
smaller squares. Birds are crossing over the area in three
directions, equally spaced, so that each of the subdivisions
is traversed by three of them. We might be tempted to
conclude that 4 x 3, or 12, would pass over the large square.
Actually there are only seven birds involved all told.
Obviously, the interceptive potential of a small square and a
larger square do not stand in the same ratio as their areas.]
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
cooerdinates 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.
[Illustration: FIG. 7. The sampling effect of a square. In
Diagram A eight evenly distributed birds are flying from
south to north, and another four are proceeding from east to
west. Three appear in each of the smaller squares. Thus, if
we were to treat any of these smaller sections as a directly
proportionate sample of the whole, we would be assuming that
3 x 16, or 48, birds had traversed the square mile--four
times the real total of 12. If we consider the paths
separately as in Diagram B, we see quite clearly what is
wrong. Every bird crosses four plots the size of the sample
and is being computed into the total over and over a
corresponding number of times. Patently, just as many
south-north birds cross the bottom tier of squares as cross
the four tiers comprising the whole area. Just as many
west-east birds traverse one side of the large square as
cross the whole square. In other words, the inclusion of
additional sections _athwart_ the direction of flight
involves the inclusion of additional birds proceeding in that
direction, while the inclusion of additional sections _along_
the direction does not. The correct ratio of the sample to
the whole would seem to be the ratio of their perimeters, in
this case the ratio of one to four. When this factor of four
is applied to the problem it proves correct: 4 x 3 (the
number of birds that have been seen in the sample square)
equals 12 (the exact number of birds that could be seen in
the square mile).]
[Illustration: FIG. 8. Rectangular samples of square areas.
In Diagram A, where as many birds are flying from west to
east as are flying from south to north, the perimeter ratio
(three to eight) correctly expresses the number of birds that
have traversed the whole area relative to the number that
have passed through the sample. But in Diagram B, where all
thirty-two birds are flying from south to north, the correct
ratio is the ratio of the base of the sample to the base of
the total area (one to four), and use of the perimeter ratio
would lead to an inaccurate result (forty-three instead of
thirty-two birds). Perimeter ratios do not correctly express
relative interceptory potential, unless the shape of the
sample is the same as the shape of the whole, or unless the
birds are flying in all directions equally.]
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
cooerdinates 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.
[Illustration: FIG. 9. The effect of vertical components in
bird flight. The four diagrams illustrate various effects
that might result if a bird with an undulating flight, such
as a Goldfinch, flew before a moon 45 deg. above the horizon. In
each case the original profile of the pathways, illustrated
against the dark background, is flattened considerably as a
result of projection. In the situation shown in Diagram A,
where the high point of the flight line, GHJ, occurs within
the field of the telescope, it is not only obvious that a
deviation is involved, but the line GJ drawn between the
entry and departure points coincides with the normal
cooerdinates of a bird proceeding on a horizontal plane. In
Diagrams B and C, one which catches an upward segment of
flight, and the other, a downward segment, the nature of the
deviation would not be detectable, and an incorrect direction
would be computed from the cooerdinates. Over a series of
observations, including many Goldfinches, one would expect a
fairly even distribution of ups and downs. Since the average
between the cooerdinate angles in Diagrams B and C, +19 deg. and
-19 deg., is the angle of the true cooerdinate, we have here a
situation where the errors tend to compensate. In Diagram D,
where the bird is so far away that several undulations are
encompassed within the diameter of the field of view, the
cooerdinate readings do not differ materially from those of a
straight line.]
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.
[Illustration: Fig. 10. The interceptory potential of
slanting lines. The diagram deals with one direction of
flight and its incidence across lines of six different
slants, lines of identical length oriented in six different
ways. Obviously, the number of birds that cross a line
depends not only on the length of the line, but also on its
slant with respect to the flight paths.]
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 deg. 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 deg. 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
cooerdinate 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.
[Illustration: FIG. 11. Theoretical possibilities of
vertical distribution. Diagram I shows the effect of a
uniform vertical distribution of birds. The figures indicate
the number of birds in the respective areas. Here the sample
triangle, ABD, contains the same number of birds as the
upright triangle, ACD, adjacent to it; the density plane may
be conceived of as a series of such alternating triangles,
equal in their content of birds. Diagram II portrays, on an
exaggerated scale, the situation when many more birds are
flying below the median altitude than above it. In contrast
to the 152 birds occurring in the triangle A'C'D', only
seventy-two are seen in the triangle A'B'D'. Obviously, the
latter triangle does not provide a representative sample of
the total number of birds intersecting the density plane.
Diagram III illustrates one method by which this difficulty
may be overcome. By lowering the line F'G' to the median
altitude of bird density, F''G'' (the elevation above which
there are just as many birds as below), we are able to
determine a rectangular panel, HIJK, whose content of birds
provides a representative sample of the vertical
distribution.]
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