Introduction

Triangulation and Trilateration is a horizontal control survey whose purpose is to determine the position of a number of control points or stations precisely. The basic framework of the control points are the triangles both in the triangulation and trilateration.

Triangulation

It is the method of providing control points by measuring all the angles of a triangle and length of a line called Base line. The length of all other sides of the triangle are computed from the measured angles and the length of base line.

Trilateration:

It is the method of establishing the control stations for providing horizontal control in which the length of all the sides of the triangle is measured. The distances are measured with Electronic Distance Measuring Instrument (EDM) with an accuracy of 1 ppm to 5 ppm.

But these days due to advent of EDM’s Trilateration is more in use as compared to triangulation.

Data required to start Triangulation

a. Azimuth of a line

b. Coordinates of a point

c. Length of base line

Objective of Triangulation and Trilateration:

1. To establish control for plane and geodetic surveys of large areas.
2. To assist in the determination of the size and shape of the earth by making observations of        latitude and departure.
3. To establish accurate control for photogrammetric surveys of large areas.
4. To determine accurate locations of points in engineering works such as:
   • Fixing centre lines and length of bridge axis of long bridges over large rivers.
   • Fixing centre lines, terminal points for long tunnels.
   • Detection of crustal movements.

Classification of Triangulation/ Types of Triangulation System:

Layout of Triangles:

1. Simple Triangles:

This is usually employed in long and narrow surveys of low precision such as for a valley or a narrow body of water.

There is only one route in this system to compute the length of unknown sides while the other systems provides at least two routes. So there is no extra check.

2. Double Chain Triangles:

This is used for covering the larger width of 2 belt of land.

3. Braced Quadrilaterals:

These are better than the chain of simple triangles. There is more than one route for computing the lengths of a side. Hence, the accuracy is increased.

4. Polygons:

In all of the above cases, we directly measure the length of a side and bearing at every 10⁰ station and compare it with the computed value.

Qualities of good figure

• The figure should be such that the computations can be done through two independent routes.
• The figure should be such that at least one (if both better) routes should be well conditioned.
• All the lines in a figure should be of comparable lengths.
• The figures should be such that least work may secure maximum progress.

Field works in Triangulation

• Reconnaissance
• Fraction of signals and towers
• Measurement of base line
• Measurement of horizontal angles.
• Measurement of vertical angles.
• Astronomical observations to determine the azimuth of the lines

 Office works in Triangulation

• Adjustment of observed angles
• Computation of lengths and sides
• Computation of the coordinates of the stations

Accuracy of Triangulation

The primary measure of the precision of triangulation is the average triangulation error i.e. the average deviations of the sum of the measured angle in the triangle form 180⁰ after correction for curvature.

In a small triangle whose sides are of the order of 2 km, the curvature of earth may be considered negligible and three measured angles should sum to 180⁰. In practice there will be a difference of a number of seconds known as Triangulation Error.

In a large triangles, error arises from the fact that, though the angles are measured in the horizontal plane, the curvature of earth causes these planes non parallel with each other. The three angles should now sum to more than 180⁰. The increment is called Spherical Excess.

Σ (Measured Angles) – ( 180⁰ + E ) = ɛ

Where,

E = Spherical excess

ɛ = Triangulation error

The spherical excess is calculated from E= A/ ( R² sinθ )

Where,

A= Area of the triangulation

R= Mean radius of the earth

Application of Triangulation / Trilateration

1. The establishment of accurately located control points for survey of large areas.
2. The accurate location of engineering works such as:
   • Center line, terminals points and shafts for long tunnels.
   • Center line and abutments for bridges of long span.
   • Complex highway interchange
3. The accurate establishing of control points in connection with the Aerial survey.
4. Measurement of deformation of structures such as dams.
5. The study of gradual and secular movements in the earth’s crust in areas subject to seismic or tectonic activities.
6. To test and construct defense and scientific facilities on high precision, engineering projects.

Advantage and Disadvantage of Trilateration

Advantage:

• Trilateration is a practical and highly accurate means of rapid control extension when properly executed, it is superior to both triangulation and traverse.
• Basic trilateration is less expensive than classical triangulation and under most conditions it is more accurate.
• In trilateration, it is not required to measure lines which all signals simultaneously in position as with triangulation.

Disadvantage:

• Trilateration has a smaller number of internal checks compared to triangulation.
• Higher order trilateration requires sampling of metrological condition to have high precision in distance measurement
• Reflector arrays have to be fixed at the station for the use of EDM. In triangulation, light signals are sufficient

Comparison of Triangulation and Trilateration:

• In triangulation, length of at least one line called the Base line is measured and rest of angles of the triangle is measured. In trilateration, lengths of all the sides of the triangles are measured directly and no angular measurements are made.
• In triangulation, lengths of all the remaining sides of the chain of triangles and hence the position of all the remaining stations are computed in terms of measured angles and measured lengths of the base line. In trilateration, lengths are measured directly.
• In triangulation, the distance are computed from the angular observations. In trilateration, angles are calculated from linear observations.
• The number of redundancies in triangulation is significantly higher than in trilateration i.e. triangulation provides more number of conditions equations or degree of freedom than in trilateration.

Example:

In trilateration: One redundancies

In triangulation: Six redundancies

Electronic Distance Measurement (EDM):

Presents days EDM’s use either infrared ( light waves) or microwaves (radio waves). The microwave system require a transmitter and receiver at both ends of the measured line, where ass the infrared systems require a transmitter at one end and reflecting prism at the other end. Microwave systems are after used in hydrographic surveying and hence have a usual upper measuring range limit of 50 km.

Infrared EDM’s come in long range ( 10 – 20 km), medium range ( 3 to 10 km) and short range ( 0.5 to 3 km)

GPS system

A network of satellite that continuously transmit coded information for locating a point an earth by measuring the distance from the satellites receiver.

GPS satellite

A group of U.S Department of Defense satellite constantly circling the earth and transmitting very low power radio signals.

Fee

No subscription fees or set up changes to use ordinary GPS.

Segment of GPS

1. Space Segments ( Satellites)

• 24 satellite ( 21 active plus 3 spares)
• Satellites about 12000 miles above the earth surface
• At least 6 of them can be received at any time
• At a speed of 7000 miles an hour and 2 times a day

2. Control Segment ( Control Station)

• Controls the GPS satellites( by tracking them and the providing them with corrected orbital and time information)
• Five control stations ( 4 unmanned monitoring station and 1 master control station)

3. User Segments

• Consists a person using GPS and GPS receiver.

Points worth Remembering

• Needs to receive at least 3 satellites for 2D.
• With 4 satellites it gives elevation too.
• Horizontal position is accurate between 7 and 15m if :
• The satellite are far enough and not in the same direction.
  • Not in the same line.
  • There is a lot of satellite received. But elevation is never reliable  ±  35m
• The antenna has to be upright if GPS has an external antenna.
• GPS is more accurate after some minutes than just after it begins to give accuracy value.

Working Principle:

• 3 satellites for 2D position(x,y)
• 4 or more satellite for 3D(x,y,z)
• GPS  receivers compares the time transmitted by satellite with its time
• Time difference tells GPS receiver how far away the satellite is

I.e. Distance = Velocity of radio wave * Travel line

Basic definition in Contouring:

• A line joining points of equal elevations is called a contour line.
• It helps to visualize the relief of ground in two dimensional plane or map.
• The method of plotting contours in a plan or map is called conturing. A contour line marked by a heavier line weight to distinguish it from intermediate contour lines is called index contour line.

An index contour line shows the elevation of nearby contour line.

Contour interval:

The difference in elevation between successive contour lines on a given map is fixed. This vertical distance between any two contour lines in a map is called the Contour Interval ( CI) of the map. The choice of a suitable contour interval in a map depends upon four principle considerations.

• Nature of the ground:

The contour interval depends upon the nature of the ground. For flat ground, a small contour interval is choosen, whereas for undulating ground, higher contour interval is chosen.

• Scale of map:

The contour interval normally varies inversely to the scale of the map i.e. if the scale of map is large, the contour interval is considered to be small and vice versa.

• Accuracy:

Accuracy needed in survey work also decides the contours interval. Surveying for detailed design work or for earthwork calculations need high accuracy so small contour interval is used. But in case of location surveys where the desired accuracy is less, higher contour interval should be used.

• Time and cost:

If the contour interval is small greater time and cost is required in survey and map plotting. So, if the available time and cost is limited larger contour interval is used.

Horizontal Equivalent:

The horizontal distance between two points on two consecutive contour lines for a given slope is called the horizontal equivalent.

Characteristics of Contour:

a. Two contour lines donot cross each other except in the case of overhanging cliff.

b. Contour overlap or unit in a single line in case of vertical cliff.

c. If the contour lines are parallel and distance apart then it shows that the ground is gentle slope.

d. If the contour line are spaced closer and non-uniform then it shows steep and undulated ground.

e. A contour line must close upon itself through not necessarily within the limits of the map.

f. A single contour line cannot split into two contour line.

g. A closed contour line with one or more higher one inside represents a hill; similarly contour line with lower value inside indicates a depression or pond.

h. Contour cross a water shead or ridge line at right angles. Thery form curves  of U-shape around it with concave side of the curve towards the higher ground. Contour cross a valley line at right angles. They form sharp curves of V-shape across it with convex side of the curve towards the higher ground.

i. Contours donot cross a river or stream.

j. Contour donot have sharp-turings.

Methods of contouring

The method of establishing/ plotting contours is a plan or map is known as conturing. In general the field methods of conturing may be divided into two classes:

a. Direct method and

b. Indirect method

a. Direct method:

Indirect method, the contour to be plotted is acutually traced on the ground. Only the required points are surveyed then plotted. In this method work has mainly two folds.

• Vertical control:

If 100m RL is to be plotted then first staff is placed on BM and then HI is to be found out, then staff point is searched on ground for the required RL of 100m as :

RL= HI – Staff reading

( Staff reading = HI – RL)

When the staff(ground) point is located by required staff reading it is pegged and number of such types of points are joined to form the contour line.

• Horizontal control:

When the required points on the ground are located then such points should be suitably controlled by the control points or lines. In small area chain survey may be used but in large area traverse may be required.

b. Indirect method:

• Square method:

This method is suitable if the are is small and ground is not very much undulating. The area to be surveyed is divided into is number of squares. The size of squares may vary from 5-20m depending upon the contour and contour interval. The RL of each square corner are determined then required contour line is interpolated.

• Cross section method:

In this method, cross-section’s are run transverse to the center line of a road, railway or canal etc. This method is most suitable for railway route survey. Spacing of the cross section depends upon the terrain of the land.

• Tacheometric method:

This method is suitable for hilly terrain. In this method contour point is located by stadia hair reading and the RL of the points are found. Instruments is places at fixed positions and reading is taken in a radial line at different angles and then RL of different points are found out.

Interpolation of contours:

It is the process of spacing the contours between the plotted points. This method is based on the assumption that the slope of ground between two points is uniform.

• Bye-eye estimation:

This method is extremely rough and is used for small scale work only. The position contour points between the guide points are located by estimation.

• By arthmetic calculation:

This method is accurate but time consuming. Contour between the guide points are located by arithmetic calculation.

• Graphical method:

In graphical method a piece of tracing paper in which several lines are drawn parallel to each other is used to located to contours.

Uses of contours map:

• Nature of ground can be understood by studying contour map. So, planning can be performed without visiting site.
• A suitable site for economical alignment can be selected for any engineering project.
• Capacity of a reservoir or the area of a catachment can be approximately computed.
• The inter visibility between two points can be determined.
• A suitable route for a given gradient can be marked on the map.
• A section of the ground surface can be drawn in any direction.
• Quality of earthwork can be computed.
• Measurement and calculation of drainage area can be done.

Principle of Electronic Distance Measurement and Use of EDM:

Electronic distance measurement(EDM) is a method of determining the length between two points using electromagnetic waves. EDM is commonly carried out with digital instruments called theodolites.

EDM instruments are highly reliable and convenient pieces of surveying equipment and can be used to measure distance of upto 100 kilometer. Each piece of EDM equipment available at Engineer supply provides dependably accurate distance measurements displayed on an easy to read digital screen.

Theodolite , Types of Theodolite and its components:

Theodolite:

It is an instrument used for measuring horizontal and vertical angles accurately. It is widely used in surveying for laying off horizontal angles, prolonging survey lines, finding difference in elevation and setting out engineering works which require higher precision.

It is used in ranging the highway, railway curve, alighnment of highway and railway, canal, tunnel, etc.

Theodolites are primarily classified as:

a. Transit theodolite

b. Non-transit theodolite

• A transit theodolite is one in which line of sight can be reversed by resolving the telescope through 180⁰ in the vertical plane. Vernier theodolite is also known and transit.

• A non-transit theodolite is one in which telescope cannot be transited.

Components of theodolite:

A transit theodolite consits of the following parts:

• Telescope
• Vertical circle
• Index frame/ T-frame/ Vernier frame
• Leavelling head ( consist of two parallel triangular plates i.e. upper tribach and lower tribach.)
• Two spindles/axes/centres
• Lower plate/ scale plate
• Upper plate/ Vernier plate
• Plate levels
• Tripod

Main parts of a theodolite:

1. Levelling head (7): Levelling head is used to attach the instrument to tripod and attach the plumb bob along the vertical axis of the instrument.
2. Lower plate/circle plate (18):  An annular horizontal plate with the graduations provided all around, from 0 to 360°, in a clockwise direction. The graduations are in degree divided in to 3 parts so that each division equals to 20 min.
   • Horizontal angles are measured with this plate.
   • The size of the theodolite is defined by the diameter of horizontal circle.
3. Upper plate (17): Horizontal plate of smaller diameter provided with two verniers. On diametrically opposite parts of its circumference. These verniers are designated as A and B. They are used to read fractions of the horizontal circle plate graduations. The verniers are graduated in 20 min and each minute is divided in 3 to 5 parts making least count  20⁰ or 10⁰ .
4. Clamps and tangent screws(15, 19): There are two clamps and associated tangent screws with the plate. These screws facilitate the motion of the instruments in horizontal plane.
5. Lower clamp screw locks or releases the lower plate. When this screw is unlocked both upper and lower plates move together. The associated lower tangent screw allows small motion of the plate in locked position.  
6. The upper clamp screw locks or releases the upper verniers plate. When this clamp is released the lower plate does not move but the upper verniers plate moves with the instrument. This causes the change in the reading. The upper tangent screw allows the fine adjustment.
7. Plate level (5):
   • Spirit level with the bubble and graduation on glass cover.
   • A single level or two levels fixed in perpendicular direction may be provided.
   • The spirit level can be adjusted with the foot screw (21) of the levelling head (7).
8. Telescope (10): The essential parts of the telescopes are eye-piece, diaphragm with cross hairs, object lens and arrangements to focus the telescope.
9. Vertical circle (1):  Circular plate supported on horizontal axis of the instrument between the A-frames. Vertical circle has graduation 0-90 in four quadrants. Vertical circle moves with the telescope when it is rotated in the vertical plane.
10. Vertical circle clamp and tangent screw (11):  Clamping the vertical circle restrict the movement of telescope in vertical plane.
11. Altitude level (2):  A highly sensitive bubble is used for levelling particularly when taking the vertical angle observations.

Adjustment of theodolites:

The adjustments of a theodolite are of two kinds:-

1.  Permanent Adjustments.

2.  Temporary Adjustments.

1. Permanent adjustments: The permanent adjustments are made to establish the relationship between the fundamental lines of the theodolite and , once made , they last for a long time. They are essential for the accuracy of observations.

The permanent adjustments in case of a transit theodolites are :-

• Adjustment of Horizontal Plate Levels. The axis of the plate levels must be perpendicular to the vertical axis.
•  Collimation Adjustment. The line of collimation should  coincide with the axis of the telescope and the axis of the objective slide and should be at right angles to the horizontal axis.
•  Horizontal axis adjustment. The horizontal axis must be perpendicular to the vertical axis.
• Adjustment of Telescope Level or the Altitude Level Plate Levels. The axis of    the telescope levels or the altitude level must be parallel to the line of collimation.
•  Vertical Circle Index Adjustment. The vertical circle verniers must read zero when the line of collimation is horizontal.

2. Temporary Adjustment: The temporary adjustments are made at each set up of the instrument before we start taking observations with the instrument. There are three temporary adjustments of a theodolite:-                 

• Centering.
• Levelling.
• Focusing.

a. Setting of theodolite over the station (Centering):

• Centering of the instrument
• Appropriate leveling with the help of tripod legs.

b. Levelling up (Levelling):

• Exact/ accurate levelling is done with the help of foot screws.

 c. Elimination of parallax:

• Focusing the eye-piece
• Focusing the objective

Uses of Theodolite-Measurement of Horizontal and Vertical Angles and Bearings

Horizontal angles: A horizontal angle is the difference in direction of two intersecting lines in a horizontal plane.

Vertical angle: A vertical angle is the difference in direction of two intersecting lines in a vertical plane. It is usually measured with respect to horizontal line and inclined line of sight at observed point. The inclined may be upwards or downwards with respect to horizontal axis.

There are three methods for angular measurements:

• Normal method
• Repetition method
• Direction method ( Reiteration method)

Components of Total station:

Total station is the latest developed surveying equipment. It is a combination of electronic tachometer and EDM instrument. It can observe distance, angles, bearings, elevation coordinate etc and displayed on its screen automatically as soon as a single key is touched.

Features of Total station:

• One Transmitting and Receiving unit ( Instrument station)
• Reflection unit ( Staff points/ target point)
• Excellent hardware feature for setup of the instrument
• Provide information with sound during field observation.
• The angular observations can be taken both in clockwise or anticlockwise direction.

Uses of Total station:

It can be used in Traversing, Trigulation, Trilateration, Trigometrical levelling and detail engineering survey like road, irrigation, hydropower, tunnel, mines, etc to measure distance angles, bearings, elevation, coordinate etc.

Electronic Data Recording:

Total station can be operated like a conventional theodolite using similar surveying procedure by the data observed by total station can be recorded automatically in the electronic field book fixed in total station. The device like Data loggers and Data recorders are used for collection and storing the observed data. Large amounts of data can be stored in this small device. The observed data are transmitted from the total station to a data recorder and these are stored together with point numbers generated by the recorder and feature codes which are entered manually on site.

So, after field observation, data collected is transferred from a data recorder to a computer through memory card for further processing,

Some common notations used in observations are,

HL = Face left horizontal angles

FR = Face right horizontal angles

V = Vertical angles

HD = Horizontal distance

VD = Vertical distance

SD = Slope distance

N = North cooridinate

E = East coordinate

Z = Z- coordinate

ANG = Angle meaasurement key

ESC =  Escape key

ENT = Enter key

F₁ – F₄ =  Function key

m =  Meter unit

F =  Inch/ Feet unit

Definition of surveying:

Surveying is the art of determining the relative positions of distinctive feature on the surface of the earth or beneath the surface of the earth, by means of measurements of distances, directions and elevations.

The branch of surveying which deals with the measurements of relative heights of different points on the surface of the earth, is known as levelling.

Primary Divisions of Surveying:

The surveying may primarily be divided into two divisions:

1. Plane Surveying
2. Geodetic Surveying

1. Plane Surveying:

The surveying in which earth surface is assumed as a plane and the curvature of the earth is ignored are known as Plane surveying.

As the plane survey extends only over small areas, the lines connecting any two points on the surface of the earth, are treated as straight lines and the angles between these lines are taken as plane angles.

Hence, in dealing with plane geometry and trigonometry is only required. Survey covering an area up to 260 sq.km may be treated as plane surveys because the difference in length between the arc and its subtended chord on the earth surface for a distance of 18.2 km, is only 0.1m.

Scope and use of plane surveying:

• It cover areas up to 260 sq.km and it is carried out for engineering projects on sufficiently large scale to determine relative positions of individual features of the earth surface.
• It is used for the lay-out of highways, railways, canals, fixing boundary pillars, construction of bridges, factories etc. The use and scope of plane survey is very wide.

2. Geodetic Surveying:

The surveys in which curvature of the earth is taken into account and higher degree of accuracy in linear as well as angular observations is achieved are known as Geodetic Surveying.

In geodetic surveying, curvature of earth’s surface is taken into account while making measurements on the earth’s surface. As the surveys extends over large areas, lines connecting any two points on the surface of the earth, are treated as arcs. For calculating their projected plan distances for the plotting on the maps, the curvature correction is applied to the measured distances. The angles between the curved lines are treated as spherical angles. A knowledge of spherical trigonometry is necessary of making measurements for the geodetic surveys.

Scope and use of Geodetic Surveying:

• It is conducted with highest degree of accuracy to provide widely spaced control points on the earth’s surface for subsequent plane surveys.

• It require the use of sophisticated instruments, accurate methods of observations and their computation with accurate adjustment.