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Section III. TRIGONOMETRIC LEVELING
This method applies the fundamentals of trigonometry to determine differences in elevation (fig. 8-7). There are two applications of this method used by the military surveyor; on long lines of sight for triangulation and electronic traverses and on short lines of sight for conventional traverses and level lines. The procedures and techniques in this chapter pertain only to the short line application. For trigonometric leveling on long lines for triangulation and electronic traverses, refer to chapters 4 and 6, TM 5—441, respectively. Trigonometric leveling is used only for lower order accuracies where the terrain is prohibitive to differential leveling or when leveling is needed in connection with triangulation and traverses.
Trigonometric leveling requires a transit, theodolite, or alidade to observe the vertical angles needed in this method. This method is particularly adaptable to uneven terrain, where level sights would be short due to the ground slopes and distance balancing, and for low order surveys where
time is a consideration. Distances should be kept below 300 meters when a stadia or standard leveling rod is used, and the curvature and refraction correction (para 8-5) is applied only if the survey accuracy requires it. Trigonometric level surveys should be tied in with sideshots to higher order elevations whenever possible.
a. The instrument is set up and leveled at a convenient location to see the starting point and the first turning point. The rod is held on the starting point. The telescope is pointed at some easily read value (a full meter) on the rod, and the vertical angle is read. The distance between the instrument and rod must be determined either by taping, by a stadia reading, or in some instances by triangulation. Now, one side and one angle of a right triangle are known (fig. 8-7). The other sides and angle can be computed. For trigonometric leveling, only the side opposite the measured angle, the difference in elevation or DE, is computed. The computation consists of multiplying the measured distance by the proper trigonometric function of the measured angle (sine; if slope distance, OR, is measured; tangent; if horizontal distance, OH, is measured). The result is the difference in elevation, DE, between the HI and the point on the rod, R. The rod reading (like in differential leveling) is added on backsights and subtracted on foresights. The computed DE is applied in the proper direction to obtain the HI or the elevation as required.
(1) Depression (minus) angle backsight (©, fig. 8-7). The rod is on a point ® below the instrument. The measured vertical angle (a) is minus or a depression angle. The measured distance is either slope (OR) or horizontal (OH). The required DE (HR) equals the distance multiplied by the sine or tangent of the angle (a). To compute the HI, the rod reading, RB, and the DE are added to the elevation of B, or HI = RB + DE + Elev. B.
(2) Depression (minus) angle foreseight (®, fig. 8-7). The rod is below the instrument, and the vertical angle is minus. The DE is computed as in (1) above. The elevation at C equals the HI minus the DE and minus the rod reading RC, or Elev. C = HI — DE — RC.
(3) Elevation (plus) angle backsight (($), fig. 8-7). The rod is above the instrument, and the vertical angle is plus. The DE is computed as in (1) above. The HI at F equals the elevation at C plus the rod reading, RC, and minus the DE,
or Hi = Elev. C + RC — DE.
(4) Elevation (plus) angle foresight (©, fig. 8-7). The rod is above the instrument and the angle is plus. The DE is again computed as in (1) above. The elevation of G equals the HI plus the DE and minus the rod reading, RG,
or Elev. G = HI + DE — RG.
b. The distance between the instrument and the stations must be known in trigonometric leveling to compute the difference in elevation. This distance may be taped, measured electronically, or read by stadia. It may be a part of another survey (such as traverse) or it may have to be measured during the leveling.
(1) Horizontal distances are simply multiplied by the tangent of the angle to get the difference in elevation. No reduction is required once the proper corrections to the measured distance are applied to get the true horizontal distance.
(2) Slope taping distances must be converted to a horizontal distance before being used in this procedure. See appendix E, for tables which list the following:
(a) Inclination corrections for a 50-meter tape (table E-2).
(6) Differences in elevation for given hori
zontal distances and gradients from 0° to 45° (table E-3).
(c) Differences in elevation for given slope distances and gradients from 0° to 45° (table E-4).
(d) Horizontal distances for given slope distances and gradients from 0° to 45° (table E-5).
(e) Differences in elevations and horizontal distances from stadia readings (table E-6).
(3) Electronic distance-measuring devices measure the slope distance between instruments. This is the straight line distance from unit to unit. If the same setup is used, and the electronic equipment is replaced with a theodolite and target or rod, the measured vertical angle can be used to convert the measured distance to a difference in elevation by multiplying by the sine of the angle.
(4) Stadia distances fall into two categories. If the instrument is level when the distance is read, the value is converted directly to a horizontal distance. When the line of sight is moved up or down from the horizontal, the rod reading cannot be converted directly, but requires an additional reduction. Stadia is more fully described in paragraph 6-26. Stadia reduction tables are provided in appendix E. The rod intercept (interval) and vertical angle are used with the tables and yield a horizontal distance and a vertical difference in elevation.
c. The vertical angle used in trigonometric leveling is the angle above or below a horizontal plane, and is designated by a plus or a minus, respectively.
(1) The transit's vertical circle is graduated from 0° to 90° on each of four quadrants. A horizontal line reads zero in either the direct or the reverse position. Vertical angle values will increase whether the telescope is elevated or depressed. The vertical angle is read on the vernier and the sign depends on the telescope position; plus if elevated, and minus if depressed.
(2) The 1-minute and 1-second theodolites use zenith distances; that is, a level sight will read 90° on the circle. As the line of sight is elevated, the value on the circle decreases, and must be subtracted from 90° to give a plus vertical angle. The depressed line of sight reading will be greater than 90° and the amount greater is the minus vertical angle. In the reversed position, the 1-minute and 1-second theodolites read 270° for a horizontal line. The amount above or below 270° is the vertical angle, plus or minus, which is used in trigonometric leveling. area. Where considerable brush cutting is necessary, additional personnel are required.When using a theodolite the zenith distance (ZD) is observed and the vertical angle (VA) computed
(3) The alidade has a stadia arc with three scales. The center scale is the angle scale and is read by vernier to the nearest minute. Its value for a level line is 30°. Vertical angles are determined by subtracting 30° from the reading. Elevation angles will give a plus result and depression angles, a minus. The other two scales marked H and V can be used in leveling. Their theory and use are described more fully in paragraph 7-3.
8-18. Limiting Factors
The chief source of error in this method is in determining the true distances between the turning points (TP's). This error can be kept to a minimum by accurately determining the stadia
contant (para 3-8c), limiting the length of the line of sight to 300 meters and the vertical angle to 4 degrees or less. Vertical angles from 6 to 9 degrees are acceptable providing the length of the sight is reduced proportionally. In no case should the vertical angle exceed 9 degrees. Other precautions to be taken are keeping the lower stadia wire at least 0.6 meter above the ground when reading stadia and using a rod level to plumb the rods.
8-19. Party Organization
The organization of the trig-level party is generally that of an instrumentman (chief of party), a recorder, and two rodmen, but this will be governed by the terrain and type of vegetation in the
The allowable error of closure will generally dictate the method and procedures to be used for lower order trigonometric level lines, and this will be specified in the project instructions.
a. Stadia Method.
(1) To run a stadia-trigonometric line of els, set up the instrument at a convenient point along the proposed line, measure and record the height of instrument (para 7-3), and take a backsight on the rod held on the bench mark. Set the
middle wire on an arbitrarily chosen graduation of the rod to be used as an index mark (usually 2.00 meters) and record the value. Then set the bottom wire on the nearest whole meter and read and record the stadia distance. Reset the middle wire on the index mark and observe and record the vertical angle (fig. 8-8). The telescope is reversed and vertical angle read again. The stadia distance should also be checked in the reverse position. The instrument is then turned in the direction of the forward rod and a foresight is taken on the selected turning point using the same instrument procedure. The instrument is moved to a new location ahead of the front turning point as in differential leveling, and the process repeated.
(2) If the line of sight from the instrument to the rbd is inclined, the stadia distance is not the true horizontal distance and must be corrected. This correction depends upon the observed stadia distance and the vertical angle. Distances should not be determined by half-stadia intervals. If it becomes necessary to do so, the separate half intervals should be observed and their sum taken as the total distance.
(3) The recorder must check the stadia and vertical readings before the instrument and rod is moved. The elevation of the forward point (TP, PBM, TBM or RM) is determined by carrying the elevation of the back point to the instrument (HI) and then to the forward point shown in figure 8-7 and described in paragraph 8-17a.
(4) Whenever the line of sight with the telescope horizontal will intersect the rod, it is preferable to read the rod as in differential leveling (para 8-13).
6. Short Base Method. When it becomes necessary to use lines of sight over 300 meters and the EDME's are not available, the short base method should be used. This method is performed in a
similar manner as the short base traverse. The short base lines must be measured to at least third order traverse accuracy with a ratio between the measured and computed lengths not to exceed one part in 50 parts. For the specifics of this method refer to paragraph 7-236, TM 5-441.
c. Electronic Measuring Methods. Electronic distance measuring may be used in conjunction with the theodolite for trigonometric leveling in the same manner as stadia distances are used in o above. Due to the increase in accuracy of the measurement of the slope distance, the length of the line of sight may be extended to meet the situation. At any time the length of the line of sight exceeds 300 meters, simultaneous reciprocal angles must be observed.
d. Alidade Method. The alidade and planetable may be used for lower order elevations. When using the alidade the vertical angle is usually measured in terms of Beamans, instead of degrees. The procedure is the same as in a above. The difference in elevation and distance determination using the scales of the stadia arc is described in paragraph 7-3. For a recording format, refer to figure 8-9, DA Form 5-72.
Section IV. BAROMETRIC LEVELING
a. On certain survey (engineer or field artillery) projects in remote and difficult terrain, the accuracy requirements for vertical control (elevations) may be lowered to such an extent that the barometric method of establishing elevations becomes the most practical and economical, particularly if modern transportation methods such as the helicopter are available. The procedures and techniques discussed in this section are based on the assumption that helicopters will be used to transport the field parties for establishing elevations by the barometric method.
6. Barometric (or altimeter) surveys are run by one of three methods—the leapfrog, the singlebase, and the two-base. The single-base method requires a minimum of observers and equipment. However, it needs a series of corrections and is neither as practical nor as accurate as the other two. The two-base method is generally accepted as the standard method for accuracy and is most widely used in engineer surveys. It requires fewer corrections than the single-base method. The leapfrog method uses the same type of corrections as
the single-base, but the altimeters are always in close relationship to each other and are operating under reasonably similar atmospheric conditions. The results are more accurate than the single-base method but less accurate than the two-base method.
Before the starting of any barometric survey and regardless of the method used, certain precautions, considerations, and corrections must be observed and/or computed.
a. Weather. Accuracy of barometric surveys is primarily related to the prevailing weather conditions (para 3-216). The weather must be favorable or the results will be inaccurate. Wind velocity is an excellent indicator of the degree of atmospheric stability and should be used as a guide. Wind velocities of 8 knots and below (Beaufort No. 3 or less, table 3-1) are considered favorable during the leveling. In wind velocities between 8 and 13 knots, extreme caution must be used. Barometric leveling should never be attempted in winds exceeding 13 knots. The early morning