1. Tachometry

2. Introduction
Tachometry is a branch of surveying where heights and distances are determined from the instrumental readings alone; these readings are usually taken with a specially adopted theodolite known as a Tachometer.

3. The field work in tachometry is rapid compared with direct leveling and measurement and it is widely used therefore to give contoured plans of areas, especially for reservoir and hydro – electric project tipping site, road and railway reconnaissance, housing sites etc. With reasonable precautions, the results of tachometry obtained can be of the same order of accuracy as, or even better than the results obtained by direct measurement in some cases.

4. In stadia tachometry, a levelling staff is held vertically at one end of the line being measured and a level or theodolite is set up above or below the other. The staff sighted and readings taken using lines engraved on the telescope diaphram as shown in figure below. The vertical angle along the line of sight can be either horizontal or inclined as shown in the below figure. The vertical compensating system of the theodolite must be in correct adjustment since vertical angles are read on face only.

6. K = the multiplying constant of instrument, usually 100
C = the additive constant of the instrument, usually 0
S = the difference between the two stadia readings
Ø = the vertical angle along the line of sight
hi = the height of trunnion axis above point p
m = the middle staff reading at X
+v = used if there is an angle of elevation
- v = used if there is an angle of depression

7. In stadia tachometry, a leveling staff is held vertically at one end of the line being measured and a level or theodolite is set up above or below the other. The staff sighted and readings taken using lines engraved on the telescope diaphram as shown in figure below. The vertical angle along the line of sight can be either horizontal or inclined as shown in the below figure. The vertical compensating system of the theodolite must be in correct adjustment since vertical angles are read on face only.

8. CON’T
H = Ks cos2 α + Ccos α (1) The vertical component of the inclined distance is V= Ks cosα sinα+ C sinα (2) V = 0.5KS sin2α + Csinα (3)

9. CON’T
which are the general equations for determining the difference in elevation between the center of the instrument and the point where the line of sight cuts the rod. To determine the difference in ground elevations, the height of instrument and the rod reading of the line of sight must be considered.
Equations (2) and (3) are known as the stadia formulas for inclined sights.

10. Example 1: A theodolite having a multiplying constant of 100 and additive constant of 0.00was centered and leveled at a height of 1.48m above a point C, of elevation 46.87m. A leveling staff was held vertically at points D and L in turn and the readings shown in table below were taken.

11. calculate; a. The levels of points D and L b
calculate; a. The levels of points D and L b. The horizontal distances; DCD and DCL

12. Example 2: The following readings were taken on a vertical staff with a tacheometer fitted with an anallatic lens and having a constant of 100.
Calculate the relative levels of the ground at A and B, and the mean slope between
the two points (A & B).

13. ERRORS IN STADIA
Many of the errors of stadia are those common to all similar operations of measuring horizontal angles and differences in elevation. Sources of error in horizontal and vertical distances computed from observed stadia intervals are as follows:
1- Stadia Interval factor is not that assumed. This condition produces a systematic error in distances, the error being proportional to that in the stadia interval factor. The case is parallel to that of the tape which is too long or too short.
2- Rod is not of standard length. If the spaces on the rod are uniformly too long or too short, a systematic error proportional to the stadia interval is produced in each distance. Errors from this source may be kept within narrow limits if the rod is standardized and corrections for erroneous length are applied to observed stadia intervals.

14. CON’T
Except for stadia surveys of more than ordinary precision, errors from this source usually are of no consequence.
3- Incorrect stadia Interval, The stadia interval varies randomly owing to the inability of the instrument operator to observe the stadia interval exactly. In a series of connected observations (such as a traverse) the error may be expected to vary as the square root of the number of sights.
4- Rod is not plumb. This condition produces a small error in the vertical angle. It also produces an appreciable error in the observed stadia interval and hence in computed distances, the error being greater for large vertical angles than for small angles. It can be eliminated by using a rod level.

15. CON’T
5- Unequal refraction. Unequal refraction of light rays in layers of air close to the earth’s surface affects the sight on the lower stadia hair more than the sight on the upper stadia hair and thus introduces systematic positive errors in stadia measurements. Whenever atmospheric conditions are unfavorable, the sights should not be taken near the bottom of the rod. 6- Errors in zenith or vertical angles. Errors in vertical circle readings are relatively unimportant in their effects on horizontal distances. For example, analysis of Eq. (7) shows that an uncertainty of 01’ in a zenith or vertical angle of 85° or 5°, respectively, yields discrepancies of m in a 100-m sight

16. CON’T
Similarly, an error of 01’ in a zenith or vertical angle of 75° or 15°, respectively, produces discrepancies of 0.02 m in a 100-m sight. With respect to differences in elevation, analysis of Eq. (8) reveals that an uncertainty of 01’ in zenith or vertical angles of 85° or 5°, respectively, results in discrepancies in elevation differences of 0.03 m for a l00-m sight. Uncertainty in the stadia interval, especially at higher angles of inclination, will have a more pronounced effect on elevation differences than errors in zenith or vertical angles.

17. Example 3
A stadia interval of m is observed with a theodolite for which the stadia interval factor is 100.0and C is m. The zenith angle is 87°20/ 30// with the middle cross hair set on 1.37 m. If the instrument has a height of instrument (h.i.) of 1.52 m above the point over which it is set and the point has an elevation of m, calculate the horizontal distance and elevation of the point sighted by the (a) exact stadia equations; (b) approximate stadia equations.

18. Example 4
The elevation of BM10 is m. the stadia interval factor is 100 and C= 0.31 m. Determine elevations for the turning points and BM10.