Tacheometry

Introduction

This method of survey consists of using either a level, theodolite or specially constructed tacheometer to make cross hair intercept readings on a levelling staff. As the angle subtended by the crosshairs is known, the distance can be calculated.

Definition

Tacheometry is an optical solution to the measurement of distance. The word is derived from the Greek Tacns, meaning 'swift', and metrot, meaning 'a measure'. Present day methods of tacheometry can be classified in one of the following three groups. The last two groups will not be covered in these notes.

  1. Stadia System: The theodolite is directed at the level staff and the distance is measured by reading the top and bottom stadia hairs on the telescope view. For further information about this process is provided in Theodolite Tacheometry
  2. Electronic Tacheometry: Uses a total station which contains an EDM, able to read distance by reflecting off a prism.
  3. Subtense Bar system: An accurate theodolite, reading to 1" of arc, is directed at a staff, two pointings being made and the small subtended angle measured.
  4. Optical Wedge system: A special theodolite with a measuring device in front of the telescope s directed at a staff. One pointing of the instrument is required for each set of readings.

Stadia System

There are two types of instruments used for stadia surveying. In the first type the distance between the two stadia hairs in the theodolite telescope is fixed. In the second type of equipment the distance between the stadia hairs is variable, being measured by means of a micrometer.

The most common method used involves the fixed hair tacheometer, or theodolite.

The notes below shows the calculation of the distance (D) from the centre of the fixed hair tacheometer to a target.

From the diagram, triangles AOB, aOb are similar

OX / = / U / = / AB
Ox / V / ab

Also if OF = f = focal length of object lens

then / 1 / + / 1 / = / 1 / (lens equation) and multiply both sides by (Uf).
U / V / f
u - / U / .f + f
V
u = / AB / .f + f
ab

AB is obtained by subtracting the reading given on the staff by the lower stadia hair from the top one and is usually denoted by s (staff intecept), and ab the distance apart of the stadia lines is denoted by i. This value i is fixed, known and constant for a particular instrument.

U = / f / .s +f
i
D = / f / .s + (f + c)
i

The reduction of this formula would be simplified considerably if the term f/i is made some convenient figure, and if the term (f + c) can be made to vanish.

D = Cs + k

In practice, the multiplicative constant generally equals 100 and the additive constant equals zero. This is certainly the case with modern instruments by may not always be so with older theodolites.

Measurements of Tacheometric Constants

The values are usually given by the makers but this is not always the case. It is sometimes necessary to measure them in an old or unfamiliar instrument. The simplest way, both for external and internal focussing instruments, is to regrad the basic formula as being a linear one of the form:

D = Cs + k

  1. On a fairly level site chain out a line 100 to 120m long, setting pegs at 25 to 30 metre intervals.
  2. Set at up at one end and determine two distances using tacheometer or theodolite, one short and one long. hence C and K may be determined.

i.e. / D1 (known) = Cs1 (known) + k
D2 (known) = Cs2 (known) + k
Distance / Readings / Intervals
Upper Stadia / Centre / Lower Stadia / Upper / Lower / Total
30.000 / 1.433 / 1.283 / 1.133 / 0.150 / 0.150 / 0.300
55.000 / 1.710 / 1.435 / 1.160 / 0.275 / 0.275 / 0.220
90.000 / 2.352 / 1.902 / 1.452 / 0.450 / 0.450 / 0.900

D =Cs + k

30.00 = 0.300 * C + k

90.00 = 0.900 * C + k

therefore C = 100 & K = 0

Any combination of equations gives the same result, showing that the telescope is anallatic over this range, to all intents and purposes.

Theodolite Tacheometry

The theory discussed so far, in The Stadia System, Measurement of Tacheometric Constants and Refraction and Curvature, all applies to the situation where the staff is held vertically and the line of sight of the telescope is horizontal. It is very seldom, however, that this situation occurs in practice. Generally a theodolite is sighted to a level staff held vertically (by use of a staff bubble), which gives rise to the situation below.

Since the staff is not at right angles to the line of sight of the instrument, the intercept cut on the staff by the stadia hairs will be too large. Let the actual distance between upper and lower stadia be s and the required projection of it at right angles to IQ be s1

\ D = Cs1 + K, but s1 = s cos q

In practice, the slope distance D is not often required. What we really want is S, the horizontal distance and V the vertical distance between the trunnion axis of the telescope and the point of the staff cut by the centre hair.

Now / S = / D cos q
= / Cs cos2q + k cos q
Also / V = / D sin q
= / Cs cos q sin q + k sin q
= / Cs sin 2q / + k sin q
2

So now the horizontal distance S = Cs cos2q + K cos q, and the vertical component is given by V = Cs cos q sin q + K sin q. In practice these can be reduced to:

S = 100 s cos2q and

V = 100 s cos q sin q

The difference in height between the two points is given by:

DH =HI + V - CL,

and the Relative Level (R.L.) of the point is given by

RL = RLA + HI + 100 s cosqsinq - CL

The use of these formulae gives the three dimensional location of the point. It is quite easy to determine the coordinates of the point if the bearing is measured as well as the staff intercepts and vertical angle, which of course is the standard field procedure.

Field Tacheometry

Tacheometric surveys are usually performed to measure the three dimensional location of points on the landscape so as to produce contour and detail plans for further work, or to produce coordinates for area and volume calculations. Observations are usually performed from known survey stations, often established by traversing. A sample of the field booking sheet is shown below:

Station: At A Date: 30/2/90 Party: CO,MS,KR Reference: Job 12/90
Instrument: T2/19 H.I.:1.49 R.l 23.45 Temp:35°C Press:1012mbar
Top / Centre / Lower / Horizontal / Vertical / Description
2.457 / 2.895 / 3.333 / 24° 27' 30" / 272° 45' 00" / Tree 2: 4,8,0.3
1.873 / 2.145 / 2.416 / 48° 34' 20" / 270° 23' 00" / Fence Corner
. . . and so on

In this example, the vertical angles have been observed by theodolite and therefore have to be coverted to an elevation for use in the formulae as they stand, or the formulae can be modified to accept zenith angles. The first set of readings will be reduced as an example.

s / = 2.457 - 3.333 = 0.876 / q = +2° 45' 00"
S / = 100 s cos 2 q = 100 (0.875) cos2 (2° 45')
= 87.40 metres (8.74)
RL / = RLA + HI + 100 s cos q sin q - CL
= 23.45 + 1.49 + 100(0.876) cos(2° 45') sin (2° 45') - 2.985
= 26.153m (26.15)

One of the most common outputs from a tacheometric survey is a plan of survey showing the features and contours. The procedure for the preparation of these will be discussed in future lectures.

The only other variable necessary to compute the coordinates of the point on which the staff was placed is of course a bearing. This is computed from the horizontal circle reading and a known or adopted reference bearing, similar to the procedure adopted when traversing.

Electronic Tacheometry

The stadia procedure is used less and less often these days, more commonly geomatic engineers use a combination theodolite-EDM known in jargon as a total station. Often these instruments are connected to a field computer which stores readings and facilitates the processing of the data electronically.

This instrumentation has facilitated the development of this method of detail and contour surveying into a very slick operation. It is now possible to produce plans of large areas that previously would have taken weeks, in a matter of days.

The maths behind the operation is very simple, it is in effect the same as the stadia formulae with the term for the distance replaced by the measured slope distance.

Visit the Equipment Database for more information regarding Total Stations

S = D cos (q)

RL = RLA + HI + D sin (q) - HT

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