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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
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
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
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
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
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
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. ,
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.
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.
