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CHAPTER XIV
TUE DETERM:INATION 017 ELEVATIONS
H. G. AVERS
U. S. Coast arid Geodetic Survey
The surface of the earth presents a varied succession of' plains7 hills
valleys and mountains with an extreme cha.n~,e in elevation, in the United
States, from 276 feet below sea-level in Death Valley to 14,496 feet above
sea-level at the top of Mt. Whitney.
The problem of' determining the relative elevations of different points
has always entered some phase of man's activities The needs of primi-
tive man were satisfied with the selection of the high and low places
by sight but even in so remote a period as that of the early Babylonians
there are evidences of' leveling by mechanical means. In the time of the
R.oma.ns the science seems to have reached a high state of development
in connection with the construction of the great highways and aqueducts.
Modern leveling methods owe their existence to the invention of the
spirit level in the latter part of' the seventeenth century. By means of
the level vial attached to a telescope it is possible to define a horizontal
line of sight with great accuracy. The horizontal line of sight used as
a reference line is the basis of all spirit level methods.
An accurate knowledge of the relative elevations of points is of the
highest importance in nearly all engineering operations and in the in-
vestigation of many physical questions. The selection of a route for a
railroad, highway, or canal depends to a large extent upon the profile
obtained by taking elevations along the proposed route. The det,ermina-
tion of the extent of the area that will be flooded by erecting a dam
across a stream requires an accurate system of levels over the adjoining
lands. Mining, land surveying, triangulation, gravity operations and
irrigation work all require data, in some form, derived from elevations.
METHOD
Leveling, as is well known, is accomplished by means of an instrument
called the level. The single instrument consists of a spirit level att.a.checl
to a telescope in such a way that the axis of the bubble is parallel to the
line of sight through the telescope. When the bubble is in the middle of
its tube the line of sight through the telescope is horizontal. The tele-
scope is free to rotate about a vertical axis and it is mounted on a portable
tripod.
212
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DETERMINATION OF ELEVATIONS
213
To determine the difference in elevation between two adjacent points,
the instrument is set up in a position from which both points can be
seen. A rod graduated in feet (or any unit of length.) is held vertically
over each point and the point on the rod which is in the line of sight
through the telescope when the bubble is in the middle of its vial is read.
In each case the rod reading gives the distance of the point below the
line of' sight and the difference of the two rod readings is the difference
in elevation, the point with the larger nod reading beings lower ifs
elevation.
NVhen the points whose difference in elevation is desired are far apart
or when their difference in elevation exceeds that obtainable from one
set-up of' the instrument due to the limits of the rod lengths, the above
process must be repeated by using intermediate points.
The rod readings on a point whose elevation is known or assumed is
called a bacl~si~,ht and the rod reading on a point whose elevation is
sought is called a foresight. In general there is only one backlight for
arty set-up of' the instrument but there may be a number of' foresights
·1.ependin, upon the number of points, whose elevations are desired,
which are within the sighting limits.
Points whose elevations are determined and which are marked in a
characteristic manner are called bench marks. Bench marks are usually
described in order that they may be recovered and used at any subsequent
period.
Art intermediate point upon which a rod is held in extending the
leveling from one bench marl; to another is called a turning point. Turn-
i.n~ points are not permanently marked and usually consist of a wooden
stake or piece of' metal driven into the ground. The rod is held on top
of' the stake which is usually pulled up when it hats served its purpose
and it is then used at another point.
The determination of the relative elevations of widely separated points
with a high degree of' accuracy requires the use of the most refined level-
ing instruments and the adoption of observing methods that will insure
minimum accumulation of' the inherent leveling errors. The instrument
and rods used for leveling of high precision in this country are described
in Chapter ~II.
Leveling on a large scale has been done by many private engineers
and corporations but hy far the most extensive projects are those a.c-
complished by government bureaus in the extension of the level control
system over the Ignited States.
Recog;nizin~ the need of uniformity in the work, the :Federal Board
of Surveys and Maps, composed of representatives from the federal ma.p
v
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214
PI CURE OF THE EAR TH
making bureaus, prepared specifications covering the various classes of
leveling that are carried on by the different bureaus.
In the specifications for vertical control, approved by the Boardj level-
ing is divided, according, to its accuracy, into four classes: vi%., first
order, second order, third order and fourth order. Leveling of' the first
order is the most accurate.
~1 ~eJ
_
CL ~SSIFIC NTION OF LEVELING
F~.rst-order leveling. :First-order leveling should be used in de-
veloping the main level net of' the United States. The lines should
be so placed that eventually no point in the country will be more
than about 50 miles from a bench mark established by leveling of
this order. All the lines should be divided into sections 1 to 2
kilometers in length, and each section should be run forward and
backward, the two runnings of' a section not to differ more than
4 mm. \/K or 0.017 foot >:llI, where K is the length of' the section
in kilometers and 2lI its length in miles.
Second-o)der level~rlg. Second-order leveling should be used in
subdividing. loops of' first-order leveling, that are too large to be
covered by unsupported third-order lines. Second-order leveling will
include lines run by first-order methods, but in only one direction,
between bench marks previously established by first-order leveling
and all double lines of' leveling whose sections, run in a backward
arid forward direction, check within the limits of 8.4 mm. ~ or
0.033 foot \/.71l, where IT is the length of the section in kilometers
and M its length in miles.
T1~ird-order levelings. Third-order leveling may be used in subd.i-
viding loops of first- or second-order leveling. so that eventually
no point in the country will be farther than 15 miles from bench
marks established by leveling, of the first, second, or third order.
Third-order lines should not be extended more than 30 miles
from lines of the first or second order; they may be sing,le-run
lines but must always be loops or circuits closed upon lines
of equal or higher order. Closing checks are not to exceed 12 mm.
by,
/h~ilometers in circuit, or 0.05 foot N/miles. in circuit.
Leveling of lower order. Leveling that allows closure checks
treater than the limit stated for third-order work, such as trigono-
metric leveling, ba.ron~etric leveling, or " flying " levels, shall be
~ o~sidered as belonging to the lower order of work. No bench marks
established by leveling that is less accurate than that of the third
order, as above described, shall be marked by standard bench-mark
tablets, except that in mountainous regions inaccessible to ordinary
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DETERMINATION OF ELEVATIONS
21a
spirit-level lines, standard marls may be used on mountain summits
to mark elevations determined by trigorrometri.c leveling; such marks
should be stamped in a distinguishing manner. Elevations inferior
to the third order in accuracy shall not be published in such a way
as to be confused with standard work of the third or higher orders.
Bench marks. All first-, second-, and third-order leveling lines
should be adequately marked, at average intervals of not more than
3 miles, preferably by metal tablets set into concrete posts, sub-
stantial buildings, outcropping rock, or large boulders. At places
from which it is likely that future leveling lines will be extended
at least three bench marls should be established within a radius
of about half a mile, but' far enough apart not to be affected by the
same disturbing causes. In addition, secondary marks on trees?
bridge seats, and similar places should be left for each mile of line.
~7 t~ J '-A
The field methods employed in first-, second- and third-order leveling,
are essentially the same except that in the fi-qst.-order leveling, and in
the single lines of second-order work, the instruments are capable of'
greater accuracy, more care is taken in the observations, and certain
corrections are applied to the results.'
In the computation of first-order leveling, corrections are applied for
curvature and refraction and for collimation where the sights are unequal
in length, for the change in rod length due to temperature variation
and for the difference between the length of the rod and the standard
length. A correction, called the orthometric correction, is also applied
to the differences in elevation due to the fact that in this work level
surfaces cannot be considered parallel surfaces.
In ordinary leveling it is sufficient to assume that all level surfaces
are parallel since the error in the results introduced by this assumption
is small compared with the errors of leveling; but where great accuracy
is desired it is necessary, especially if the level line is at a high altitude
and runs in a north and south direction, to take account of the fact that
level surfaces converge as the poles of the earth are approached.
All points on a level surface have the same potential and this potential
is a function.of the product of gravity at the point and the height of the
point above the sea-level 'surface. Since gravity is least at the earth's
equator and increases as one goes north or south, reaching, a maximun~
at the poles, the distance between surfaces of different potential must
decrease as the force of gravity increases. A level surface 1,000 meters
above sea-level at the equator is only 995 meters, approximately, above
sea-level at the poles. That is, if it. were possible to run a line of levels
along a perfect level surface from the equator to the poles the results
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ClO
PI CURE OF THE EAR TH
of the leveling would show no change in elevation although the level
surface at the poles would be nearer the sea-level surface. It is therefore
necessary to correct the results of leveling so that the resulting elevations
of points represent the vertical distances of these points above the refer-
ence datum. In running, a line of levels from south to north in the north-
ern hemisphere the resulting elevations are too high. A line of' levels
running due east and west is not affected byT the convergence of the level
surfaces, because in an east and west direction the level surfaces are
Parallel.
THE FUNDAMENTAL LEVEL NET
The various lines of leveling, of' high precision (first-order) form what
is known as the fundamental level net. Over 03,000 miles of' leveling,
of' high precision have been run by the U. S. Coast and Geodetic Survey,
the IJ. S. Geological Survey, the Corps of' Engineers, U. S. Army, the
Mississippi River Commission, the Missouri River Commission, the
IJ. S. Lake Survey, the Pennsylvania Railroad, the Baltimore and Ohio
Railroad, and the BuiTalo, Rochester and Pittsburgh Railroad. This
vast network of leveling enters all the states of the Union except one,
and forms 135 circuits.
Most of' this leveling has followed the routes of railroads but since
the improvement of the highways the lines have been extended over these
routes also. This work is perpetuated by bench marks consisting of'
metal tablets, bearing an appropriate legend, set in buildings, masonry
structures, and in the top of' stone or concrete posts. The bench marks
about 2: miles apart on an average, are located in towns, at road crossings
and in other places where they can be conveniently reached by surveyors
or engineers who have need of them.
The first leveling in the LTnited States, now classed as leveling of high
precision, was run in 1879 when a line of levels was started at Sandy
Hook, N. J., to provide elevations for the transcontinental arc of triangu-
lation. As the number of lines in the country increased and formed
circuits, it became necessary, in order to distribute the closing errors
of' the circuits, to adjust the results by the method of least squares.
There have been four general adjustments of the fundamental level
net of the United States, each succeeding adjustment having become
necessary by the addition of new lines to the net. The fourth and last
adjustment was made in 1912, the net at that tinge being composed of
about 29,000 miles of leveling.
The successive adjustments necessarily gave different values for the
elevations of' the bench marks, and the changes in certain sections of' the
country were of' such sizes that the older values could not be held. The
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DETERMINATION OF ELEVATIONS
217
last adjustment shows that the net is sufficiently extended and the level-
ing of such strength and accuracy that the elevations can be considered
as standard. So far as surveying, and engineering purposes are concerned,
they may be held for an indefinite period.
The lines run since the last adjustment was made have been fitted
to the net without in any way disturbing the previously adopted elevations.
From time to time, in the future, general adjustments of the level net
will no doubt be made in order to obtain the theoretically best elevations
of the junction points; but such adjustments will not disturb the standard
elevations, unless they are found to be appreciably in error on account
of blunders in the leveling or because of the disturbance of the marks.
DATUM
Bel:'ore art elevation can be assigned to a bench mark it is necessary
to adopt a reference plane or datum. For local leveling projects, where
only relative elevations are used, this is usually any convenient arbitrary
plane. In case the elevations are to serve as the basis of other leveling
operations, and where leveling projects overlap, it is desirable for the
proper coordination and comparison of: the elevations that they be referred
to a common datum.
The elevations in the fundamental level net are referred to mean sea-
level as a datum. (See chapter on Mean Sea-Level.) Mean sea-level
provides a convenient datum that can be established easily at places
along the sea coast. \Vith such a datum, lines of levels can be started
at widely separated points along the coast and carried into the interior
of' the country without waiting for the development of' the entire level
net to furnish the elevations above the common datum.
For datum purposes mean sea-level is assumed to be at the same eleva-
tion on the open coasts of the Atlantic Oeean, the Gulf of Mexico and
the Pacific Oeean.
The fundamental level net has been connected with mean sea-level at
Portland, Me., Boston, Mass., Ft. Hamilton, N. Y., Sandy Hook and
Atlantic City, N. J., Old Point Comfort and Norfolk, Va., Brunswick,
Gal, :Fernandina, St. Augustine and Cedar Keys, Fla., Biloxi, Miss.,
Galveston, Tex., San Diego, San Pedro, and San Franeisco, Calif., Ft.
Stevens, Oreg., and Seattle, Wash.
ACCURACY
The aeeuraey of' leveling of high precision is best shown by the closing
errors of the circuits in the fundamental level net. The correction neces-
sary to close the circuits is at the rate of about 0.00063 foot per mile
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218
FI CURE OF THE EAR TH
on an average. In a hundred miles this rate ok closure is equivalent
to about 3- inch. That is, if one starts at a certain point, levels around a
circuit 100 miles in length returning to the starting point, the results
of the leveling would cheek the elevation of the starting point by 3- inch
on an average.
The above figures should not be taken as representing the ultimate
accuracy obtainable with the instruments and methods used in leveling,
of high precision. In the extension of the leveling, the observations are
not made with a view to obtaining the highest possible accuracy but
only to insure that the accuracy will be such that the leveling will fall
within the limits set for this class of work.
One of the most accurate portions of the fundamental level net is that
which covers the New England states. The 2,a=06 miles of leveling in
that area form eight complete land circuits in which the average closure
is at the rate of 0.00036 foot per mile. The closing error of the largest
circuit, 575 miles in circumference, is only 0.0663 foot or at the rate of
about 0.00011 foot per mile.
VARIATION OF MEAN SEA-LEVF,L FROM A LEVEL SURFACE
Although for datum purposes it is assumed that mean sea-level is
everywhere at the same elevation on the open coasts, the results of leveling,
between adjacent tidal stations have in a number of instances shown a
larger difference between the elevation of the mean sea-level planes than
earl be attributed to the systematic or accidental errors of the leveling
observations.
AYhen the tidal stations on the Atlantic coast at Fort Hamilton, N. Y.,
and Portland, Me., were connected by a line of high-precision leveling
the results showed that the plane of mean sea-level at Portland was 169.4
millimeters higher than the plane of mean sea-level at Et. Hamilton.
Similarly, on the PaciDe coast, the leveling showed the plane of mean
sea-level at Vancouver, B. C., to be 102.8 millimeters higher than the
same plane at Seattle, Wash.
These differences are about three times as large as the closures obtained,
on an average, for complete leveling circuits. This, together with the
feet that the plane of mean sea-level at the northern station was in each
ease the higher, led the U. S. Coast and Geodetic Survey to make a
study involving all the tidal connections along, the coast.
For the purpose of this study a special adjustment of the fundamental
level net was made. Only the more modern lines of leveling, were included
ifs the adjustment, their length totalin=, 40,000 miles. These lines formed
104 circuits.
,
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DETERMINATION OF ELEVATIONS
219
In the adjustment, which was made by the method of conditions, each
circuit closure requiring an equation, the entire net was allowed to swing
free on one mean sea-level connection, that at Galveston, Tex. The eleva-
tion of the plane of mean sea-level at the other tidal stations was then
computed and referred to the plane at Galveston as zero.
Starting at Galveston and proceeding, eastward along the coast. of the
Gulf of Mexico, the planes of mean sea-level at Biloxi, Miss., :Pensacola
and Cedar I(e.vs, Fla., are respectively 0.07 meter, 0.02 meter and 0.13
meter lower than the plane of mean seat-level at Galveston.
Starting at St. Augustine, Fla.., where mean sea-level is 0.21 meter
lower than at Galveston, and proceeding northward along the Atlantic
coast, the elevation of mean sea-level at the following, places is: Fernan-
clina, :Fla.., - ().19 meter; Brunswick, Gal, - 0.16 meter; Norfolk, Va.,
-0.16 meter; Cape Stay, N. J., 0.06 meter; Atlantic City, N. J.,
-0.05 meter; Fort Hamilton., N. Y., - 0.05 meter; Boston, Mass., + 0.01
Dieter; and Portland, Me., +0.07 meter.
On the Pacific Ocean, starting at San D;e~,o, Calif., where mean sea-
level is 0.40 meter higher than at Galveston, and proceeding, northward
the elevation of mean sea-level at the following places is: San Peclro,
Calif., +0.32 meter; San Francisco, Calif., +0.44 meter; Fort Stevens,
Oreo., +0.79 meter; Seattle, Wash., +0.66 meter; and Anacortes, mash.,
+0.6a~ meter.
These values seem to indicate that along the coast of the Gulf of
Mexico there is a general slope downward in the mean sea-level surface
front west to east, and that along the Atlantic and Pacific coasts there
is a general slope upward from south to north.
Comparing mean sea-level on the Pacific coast with mean sea-level of
the Atlantic coast, it is found that at San Diego, Calif., it is 0.~9 meter
higher than at Fernandina, Fla.; at San Francisco,. Calif., it is 0.49
meter higher than at Atlantic City, N. J., and at Seattle, Slash., it is
0.59 meter higher than at Portland, Me.
~ 1 1 · · ~ ~ · ~
The leveling thus indicates
tnat In approximately tne same latitude mean sea-level on the Pacific
coast is higher than on the Atlantic coast. This result agrees in sign
with that shown by the leveling, across the Isthmus of Panama, where it
was found that mean sea-level on the Pacific- coast is approximately 0.2
meter higher than on the Atlantic coast.
The slope of the mean seat-level surface is not peculiar to the coasts
of the United States alone. A rise of mean sea-level from south to north
on the east and west coasts of England and Scotland has been noticed
and the leveling across France shows the mean level of the Mediterranean
Sea to be lower than that of the Atlantic Ocean at Brest.
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220
FIGURE OF THE EARTH
This phenomenon may be due to the differences in barometric pressure,
prevailing winds, the salinity of the sea, and to other factors that tend
to distort the sea-level surface; but studies conducted along these lines
have not as yet produced conclusive results.
ELEVATIONS OF MOUNTAIN PEAKS
The elevations of mountain peaks are not as a rule determined by
spirit leveling, although the elevations of a few, notably, Mt. Whitney,
Pikes Peak, and Mt. Washington, have been determined by this method.
Mountain peaks do not in general have a trail leading to the top over
which spirit levels can be run, and then, too, there are easier and less
expensive methods for determining their elevations with an accuracy
sufficient to make them valuable for many purposes.
One of the most common methods for determining the elevations of
mountain peaks is that of leveling by vertical angles, or, as it is usually
called, trigonometric leveli.n~. This method consists of measuring the
angle of elevation or depression of the peak at a station whose elevation
is known and is carried on in connection with triangulation. The distance
of the peak from the known station is obtained from the triangulation
and may be as great as 10~0 miles. The elevation of the peak could then
be computed with great accuracy were it not for the uncertain value
of the refraction of the line of sight from the peak to the station. This
refraction is very variable, having different values at different hours of
the day, on different days, and in different seasons. Because the physical
condition of the atmosphere cannot be known over long lines of sight
the accuracy of trigonometric leveling, falls considerably below that of
spirit levelin,. Tests of the accuracy of trigonometric leveling, indicate
that the difference of elevation of two points by this method is correct
within one or two inches to the mile when vertical angles. have been.
Observed in both directions over the line joining the two points.
REFERENCES
Molitor, Da\id. The theory and practice of precise spirit leveling. Trans. Am.
Soc. Civ. Eng., June 1901.
Jordan, W. Handbuch der Vermessungskunde.
Helmert, F. R. Die mathematischen und physikalischen Theorieen der hoheren
Geodasie.
U. S. Coast and Geodetic Survey Report for 1893, Part II.
U. S. Coast and Geodetic Survey Spec. Pub. Nos. 18, 22 and 140.
Ordnance Survey, The Second Geodetic Leveling of England and NArales, 1912-1921.
Lallemand, Charles. Nivellement de haute precision.
Representative terms from entire chapter:
bench marks
