After reading this article you will learn about: 1. Definition of Tacheometry 2. Instruments used in Tacheometry 3. Methods 4. Stadia System 5. Tangential System 6. Field-Work.
Definition of Tacheometry:
Tacheometry is a branch of surveying in which the horizontal and vertical distances are determined by angular observations with a tacheometer, the chaining operation being altogether eliminated Tacheometry is not as accurate as is changing, but it is far more rapid in rough and difficult country where levelling is tedious and chaining is both inaccurate and slow.
Thus it is best suited when obstacles such as steep and broken ground, deep ravines, stretches of water or swamps are met with Tacheometry is mainly used while preparing contour plans and traversing and is also suitable for hydrographic surveys, location surveys of roads, railways, etc. It is also sometimes employed for small surveys in which elevations are not determined.
Instruments used in Tacheometry:
The main instruments used in tacheometry are:
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(i) A tacheometer, and
(ii) A levelling or stadia rod.
(i) Tacheometry:
A taceometer in general sense is a transit theodolite having a telescope fitted with a stadia diagram, i.e. a telescope equipped with two horizontal hairs called stadia hairs in addition to the usual central hair. The additional hairs are equipped from the central one and are also known as stadia lines. The types of stadia diagram commonly used are shown in fig. 10.1.
The kinds of telescopes used in stadia surveying are:
(a) The external focussing,
(b) The internal focussing, and
(c) The external focussing fitted with an anallatic lens.
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The term tacheometer is restricted to a transit theodolite provided with an anallatic telescope.
The essential characteristics of a tacheometer are:
(a) The value of the multiplying constant f/i should be 100.
(b) The telescope should be powerful having magnification20 to 30 diameters.
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(c) The aperture of the objective should be large about 4 cm in diagram in order to have a sufficiently bright image.
(d) The eye — piece should be of greater magnifying power to render clear staff reading even from a long distance.
(ii) Stadia Rod:
An ordinary levelling staff can be used if the sights are short but in tacheometry since the sights are usually of much greater length, therefore, an ordinary levelling staff cannot serve the purpose. But a specially designed graduated rod known as stadia rod is used.
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The stadia rod is transport, it may be folding or telescopic. It is 3 to 4m long and 5cm to 15 cm wide. The graduations are bold and clear with a least count usually less than the least count of an ordinary levelling staff, the stadia rods should be as light as possible. Some common patterns of stadia rods are shown in fig. 10.2.
Methods of Tacheometry:
The underlying principle common to different methods of tacheometry is that the horizontal distance between an instrument station and a point as well as the elevation of the point relatively to the instrument can be determined from the angle subtended at the instrument station by a known distance at point and the vertical angle from instrument to the point.
The various tacheometric methods employ the principle in different ways and differ one another in methods of observation and reduction, but may be classified under two heads:
(i) The stadia system.
(ii) The tangential system.
In the stadia system, the observation are taken with the stadia wires of the tacheometer and in the tangential system the angles of elevation are measured from instrument station to the points with a theodolite and their tangents are used to determine the horizontal of the telescope for necessary but the stadia system needs only one and is more commonly used.
The Stadia System of Tacheometry:
In the stadia system of tacheometry there are two methods of surveying viz:
(i) Fixed hair method, and
(ii) Moveable hair method.
(i) Fixed Hair Method:
In this method, the distance between the stadia hairs is fixed. When a staff is sighted through the telescope, its certain length is intercepted by the stadia hair and from tins value of staff intercept, the distance from the instrument to the staff can be determined. The staff intercept varies with its distance from the instrument. This method is most commonly used in tacheometry.
(ii) Moveable Hair Method:
In this methods, the stadia hair are not fixed but can be moved by means of micrometer screws. The stall is provided with two vanes or targets fixed at a known distance apart. The variable stadia distance is measured, and from this value the required horizontal distance may be found out. The method is now rarely used.
General Principle of Stadia Tacheometry:
The principle of stadia taceometry is explained as follows: (Fig.10.3)
Let O= the optical center of the object – glass
a, b and c= the bottom , top and central hairs at diagram
A,B and C= the points on the staff cut by the three lines
a b= i the interval between stadia lines.
(ab is the length of the image of AB)
AB=S=the staff intercept (the differences of the stadia hair readings)
f= the focal length of object – glass i.e, the distance between the centre (O) to the principal focus (FG) of the lens.
u – the horizontal distance from the optical centre (O) to the staff.
v = the horizontal distance from the optical centre (O) to the image of the staff, u and v being called the conjugate focal length of the lens.
d = the horizontal distance form optical centre (O) to the vertical axis of the taceometer.
D = the horizontal distance from the vertical axis of the instrument to the staff,
The constant f/i is called the multiple constant and its value is usually 100, while the constant (f+d) is called the additive constant and its value varies from 30 cm to 60 cm in case of external focussing telescope, it is very small varying from 10 cm to 20 cm and is therefore oftenly ignored.
To make the value of additive constant zero, an additional convex lens, known as anallatic lens, is provided in the telescope between the object – glass and eye piece at a fixed distance from former. By this arrangement, calculation work is reduced considerably.
The equation 10.1 is applicable only when the line of sight is horizontal and the staff is held vertical.
Determination of Stadia Constants of a Tacheometer:
There are two methods available for finding the values of the stadia constant f/i and f + d of a given instrument.
First method:
In this method, the values of the constants are obtained by the computations form the field measurements.
Procedure:
(i) Measure accurately a line OA about 300 m long, on a fairly level ground and fix pegs along it at intervals of say, 30m.
(ii) Set up the instrument at O and obtained the staff intercepts by taking stadis reading on the staff held vertically at each of the pegs.
On substituting the values of D and S in the Equations 10.1, we get a number of equations which when solved in pairs, give the several values of the constants:
their mean value being adopted to the values of the constants. Thus, if D1,D2, D3, etc.=the distances measured from the instrument, and S1, S2, S3 etc.= the corresponding staff intercepts.
Then we have:
Second Method:
In this method, the value of the multiplying constant f/i is found by computations from the field measurements and that of the additive constant (f+) is obtained by the direct measurements at the telescope.
Procedure:
(i) Sight any far distant – object and focus it.
(ii) Measure accurately the distance along the top of the telescope between the object -Glass and the plane of the cross -hairs (diagram screw) with a rule, the measured distance being equal to the focal length (f) of the objective.
(iii) Measure the distance (d) from the object— glass to the vertical axis of the instrument.
(iv) Measure several lengths D1, D2, D3 etc. along OA from the instrument – position O and obtained the staff intercepts S1, S2, S3 ate. at each of these lengths.
(v) Add f and d find the values of the additive constant (f+d).
(vi) Knowing (f+d), determined the several values of f/i from the equation 10. 1.
(vii) The mean of the several values gives the required value of the multiple constant f/i. Calculation work is much simplified, of the instrument is placed at a distance of (f+d) beyond the beginning O of the line.
Note:
The value of the additive constant in case of an internal focussing telescope cannot be determined in this way. One has to depend upon the value supplied by the maker.
Theory of Anallatic Lens:
An additional convex lens, called an anallatic lens, is provided in the external focussing telescope between the eye — piece and the object — glass at a fixed distance from the later, to eliminate the additive constant, (f+d), from the distance formula:
in order to simplify the calculation work. The anallatic lens is seldom placed in the internal focussing telescope since the value of the additive constant is only a few centimeters and can be neglected. The disadvantage of the anallatic a lens is the reduction in brilliancy of the image due to increase observation of light.
The value of the additive constant, (δ+d) can be made equal to zero by bringing the apex (G) of the taceometric angle AGB (Fig. 10. 4) into exact coincidence with the centre on\f the insrument.
The theory of anallatic lens is explained ad follows:
In fig. 10.4:
Let, S = the staff intercept AB.
i = the length b a of the image of AB i.e. the actual stadia interval when the anallatic lens is interposed.
i = the length ba of the image of AB when no anallatic lens was provided.
O = the optical centre of the object – glass.
O = the optical centre of the anallatic lens
e = the distance between the optical centre of the object glass and the anallatic lens.
f = to cel length of object glass.
f’ = focal length of the anallatic lens.
F = Principle focus of the anallatic lens.
G = the centre of the instrument.
d = the distance from the centre of the object — glass top the vertical axis of the instrument.
D – the distance from the vertical axis of the instrument to the staff.
f1and f2 = the conjugate focal length of the object —glass.
k = the distance from the optical centre of the object glass to the actual image b a.
(k— e) and (f2 —e) = the conjugate focal length of the anallatic lens.
The ray of light from A and B are refracted by the object — glass to meet at F. The anallatic lens is so placed that F is its principal focus. Thus ray of light would become parallel to the axis of the telescope after passing through the anallatic lens and give actual image b a of the staff intercept AB.
The negative sign is used in (ii) since b ‘a’ and ba are on the same side of the anallatic lengs.
now the conditions that D should be proportional to S requires that the 2nd and 3rd terms in (v) are equal to zero so that
In this condition, the apex G of the tacheomeric angle AGB exactly coincides with the instruments
Reduction of Readings:
In practice, the horizontal and vertical distances are not calculated by the direct application of formulae, since it is much laborious.
But they are found by the following means which are also based on these formulae:
(i) Taceometric tables.
(ii) Stadia diagrams.
(iii) Stadia slide rule.
The calculation work is also much reduced by the use of direct reading tachemetres.
(i) Tacheometric Tables:
There are various forms of taceometric tables published by different authorities. The tacheometric tables which are in common use have been at the end of the book as Appendix I. They provide horizontal and vertical distances for one metre of the staff intercept when the multiplying constant of the instrument = 100 and the additive constant = 0.
The modern taceometre which are fitted with the anallatic lens give these values of the constants, the horizontal distance for 1m staff intercept;
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and vertical distances for 1m staff intercept
The tables provided these values for different values of varying from 0° to 30°
For example, suppose, the vertical angle is 3° 20^ and the staff intercept is 1.70m. From the tables, it is seen that horizontal and vertical distance for 1 metre staff in percept i.e.
Thus for 1.70m staff intercept, the horizontal distance = 1.70 x 99 .67 – 169.439 m and the vertical distance = 1.70 x 5.80 = 9.86 m.
(ii) Stadia Diagrams:
The stadia diagrams show graphically the quantities
The diagram are available in different forms but surveyors often prepare these diagrams of their own design. The use of stadia diagram is consider faster than the use of tables but can be used for ordinary distance.
(iii) Stadia Slide Rule:
The horizontal and vertical distances are computed conveniently by stadia slide rule. Stadia slide rules are available in different patterns but the one in common use is constructed like the ordinary slide rule, except that on the slide rule are given values of cos2 and 1/2 sin 2 , these qualities being plotted to a log scale. The stadia slide rule is equally suitable for the field or office work.
The Tangential System of Tacheometry:
This method is used when the telescope is not fitted with a stadia diagram. In this method, the telescope is directed towards the staff to which the horizontal and vertical distances are to be measured and two vertical angles to two vanes or targets on the staff at a known distance (S) apart are taken.
The horizontal and vertical distances are then calculated as follows:
Case 1:
When both the observed angles are angles of elevation: (Fig. 10.10).
In fig , 10.10 let
O=The instruments station.
O’=The position of the instruments axis
P= The staff station.
∠AO’Q= θ1= vertical angle to the upper vane.
∠BO’Q= θ2 =” “ “ lower
AB=S=the staff intercept.
BQ=V= the horizontal distance from the inst, axis to the lower vane.
O’Q=D= the horizontal distance from the inst. station O to the staff station P
PB= h= the height of the lower vane above the foot of the staff.
Case II:
When both the observed angles are angles of depression: (Fig. 10.11).
Case III.
When one of the observed angle is the angle of elevation and the other an angle of depression:
Subtense Bar Measurements:
A subtense bar (fig. 10.17) is a horizontal staff with targets fixed at a known distance apart. It is about 4m long having a small spirit level and a quick levelling head.
A sight rule, provided at its centre, can be placed along the line of sight by viewing the telescope of the theodolite thought the vanes. The bar is mounted on a tripod and is placed at right angles to the line of sight for making observations. After levelling and aligning, it is clamped by means of clamp screw.
The targets, made of discs of about 20 cm diameter are painted red on one side, and white on the other. The centres of body the sides of the targets are painted black in 7.5 cm diameter. The targets are placed at a distance of 2.5 m and 3 m. When the targets are placed 2.5 m apart, the white faces are to face the instrument and when they are placed 3m apart, the red faces face the instrument.
The horizontal and vertical angles are measured with a transit theodolite. For measuring vertical angles the method will be similarly to the movable hair method of stadia tacheometry and the distances are similarly deduced. For measuring horizontal angles, subtended at the instrument station by the two targets, the method of repetition is used, the horizontal distance.
D between the instrument station and the subtense bar station is found as follows:
Let O= the position of the instrument for measuring the horizontal angle 0 by the horizontal circle of theodolite.
AB= horizontal base of a length S.
Levelling by Stadia:
Levelling by stadia tacheometry is an indirect and rapid method of levelling. It is suitable where country is rough and the precision needed is not great. The transit should preferably be provided with a sensitive control level for the vertical vernier so that the error may be readily eliminated.
The method of running a line of levels by this method is as follows:
(i) Set up the transit at a convenient point.
(ii) Take a back sight on the staff held at a B.M., first by observing the stadia interval and then by measuring the vertical angle to some arbitrarily chosen mark on the staff.
(iii) Establish a change point in advance of the transit, and take similar observations, the vertical angle being measured with horizontal cross-hari set on the same mark as before.
(iv) Move the trait to a new point in advance of the change point and repeat the process.
(v) Record the stadia distance and vertical angles and also the staff reading which is used as an index when vertical angles are measured.
Note:
For any set up, the difference in elevation determined from either the back sight or fore sight observation is the difference in elevation between the index mark on the staff and the centre of the instrument. And the algebraic sum for the back sight and fore sight is the total difference is elevation between the two positions of the index mark.
Field-Work in Tacheometric Survey:
1. Suitability:
Tacheometric survey is mainly suitable for contouring because simultaneous calculations for horizontal distances and differences of level are possible from the same set of the observed values. It is particularly useful in hilly areas where the slopes are steep and country is rough and therefore ordinary levelling and chaining become rather difficult. It is also suitable for carrying out traverses and filling in detail in rough and rugged terrain where measurement of distances by chain is not easy.
A tacheometric survey is conducted by running a closed or open traverse depending upon the area to be survey and locating the required detail from the traverse stations. Tacheometric stations should be so selected that they will command a clear view of the area to be surveyed and that the large vertical angles are avoided.
2. Equipment:
(i) A tacheometer,
(ii) A stadia rod or levelling staff,
(iii) A tape,
(iv) Ranging rods,
(v) Pegs etc.
3. Field-Party:
The field-party consists of:
(i) The surveyor who is in-charge of the party and supervises all operations such as reconnaissance, selection and location of stations, positions of staff men, etc.
(ii) The observer who is responsible for actual observations.
(iii) The recorder who records the observations in the field-book and assists the observer.
(iv) Three or more staff men.
(v) Two or more axe-men for clearing and fixing pegs etc.
4. Procedure:
The tacheometric survey should be conducted in the following steps:
(i) Set up the instrument over the station-point. Centre and level it accurately.
(ii) Set up vertical vernior to zero and measure the height of the instrument i.e. height from top of peg to centre of the object-glass with a tape or stadia rod through the object-glass.
(iii) Orient the instrument correctly at the first station of the traverse either with reference to the true meridian or with reference to the magnetic meridian.
(iv) Sight the staff held on the nearby bench mark and observes the vertical angle, the bearing and the readings of the three hairs. If there is no bench mark nearby, flying levelling may be done from any B.M. to establish another one near the area of the survey.
(v) Sight all the representative points around the station and within the range of the instrument and observes at each the bearing, the vertical angle and the staff readings at the three wires, the bearing being taken to the nearest 5′ and the vertical angle to the nearest 1′.
(vi) Take a fore sight on the second traverse-station and observe the bearing, the vertical angle and the staff readings of three wires.
(vii) Shift the instrument and set it up at the second station.
(viii) Measure the height of instrument as before.
(ix) Back sight the staff held on the first station and observes the bearing, the vertical angle and the staff readings of three wires.
(x) Sight all the points around the second station and within the range of the instrument as described above.
(xi) Take a fore sight on the third station and take necessary observations.
(xii) Proceed similarly at each of the successive points.
Note:
Since each station is sighted twice, the two values for the distance and elevation are obtained. If they agree within the limits of accuracy, the average of the two values may be taken and if not work should be repeated.
5. Field-Book:
The field-notes are recorded in the form given at the next page as table 10.1.
Errors in Stadia Surveying:
The sources of errors in stadia measurements are as follows:
1. Instrumental Errors.
2. Personal Errors.
3. Natural Errors.
1. Instrumental Errors:
(i) Imperfect adjustment of the tacheometer:
This error can be eliminated by carefully adjusting the instrument, particularly the altitude bubble.
(ii) Incorrect divisions on the stadia rod:
In ordinary work, this error is negligible but for precise work, the error can be minimised by using the standardised rod and applying corrections for incorrect length to the observed stadia intervals.
(iii) Incorrect value of the multiplying constant (f/t):
This is the most serious source of error. The value of the multiplying constant should be tested before commencing the work by comparing the stadia distances with measured distances during the hours which correspond to those of field-observations.
2. Personal Errors:
(i) Inaccurate centering and levelling of the instrument.
(ii) Non-vertical by of the staff or rod. It may be eliminated by using a plumb-line or a small circular spirit level with the staff.
(iii) Inaccurate Focussing.
(iv) Reading with wrong hair.
The personal errors can be eliminated by applying habitual checks.
3. Natural Errors:
(i) High wind:
The work should be suspended in high wind.
(ii) Unequal refractions:
This error is prominent during bright sunshine and mid-day hours of hot summer days. The work can be suspended under such circumstances.
(iii) Unequal expansion:
The instrument should be protected by an umbrella during hot sun.
(iv) Bad visibility:
It is caused by glaring of strong light coming from the wrong direction.
Degree of Accuracy:
The error in a single horizontal distance should not exceed 1 in 500, and in a single vertical distance 0.1 m.
Average error in distance varies from the 1 m 600 to 1 in 850.
Error of closure in elevation varies from 0.08 √km to 0.25 √km where km = distance in km. error of closure in a stadia traverse should not exceed 0.055 √P metres, where P = perimeter of the traverse in metres.