Transcript SurveyingIIIx

SURVEYING III
NATIONAL GEODETIC
NETWORK
Juliana Useya
Department of Geoinformatics and Surveying
COURSE CONTENT
Geodetic networks of Zimbabwe
 Network densification
 Classical geodetic positioning techniques
triangulation,
trilateration,
 traversing,





Design, reconnaissance and establishment of 2D networks,
monumentation and signaling
Calibration and field procedures
Station adjustment
National height networks
 Geodetic levelling datums
 Triangulated heights
Transformations and coordinate systems
 The arc datum and its transformation
 Gauss and UTM coordinates
2
What is the
meaning of
“…work from the
whole to the
part…”
3
OVERVIEW OF TODAY’S LECTURE









Definition of geodetic network
Purpose of geodetic network
Methods of formation: Advantages and disadvantages
Triangulation principle
Types of triangulation
Classification of triangulation
Triangulation figures and layouts
Selection criteria of layouts
Well conditioned triangles and strength of a figure
4
NATIONAL GEODETIC NETWORKS



A geodetic network is a network of triangles which are
measured exactly by techniques of terrestrial surveying or
by satellite geodesy.
National geodetic network serves the mapping control of a
geopolitical area e.g. country-level.
Purpose
Provide framework
(basic control network) for
topographical, cadastral, engineering surveys, etc.
 Monitor crustal movement.
 Using observations of astronomical latitude, longitude and
gravity to measure the size and shape of the earth and its
gravity field.

5
METHODS OF FORMATION

Triangulation- is the process of determining the location of a point
by measuring angles to it from known points at either end of a fixed
baseline.
OR is method of surveying based on the trigonometric proposition that if
one side and two internal angles are known, the remaining sides can be
computed.

Precise traversing - is theodolite traversing using more refined

Trilateration – is a method of control extension, control breakdown

Radio and electronic satellite positioning
instruments and devices for linear and angular measurements.
and control densification that employs electronic-measuring instruments
to measure the lengths of triangle sides (rather than horizontal angles as
in triangulation).
(e.g. Doppler and GNSS ).
6
ADVANTAGES


Triangulation
 Only one distance measurement in the entire network.
 Has a higher number of internal checks compared to
classical trilateration.
Precise traversing
 Less reconnaissance and organization needed.
 Traverse can take any shape and thus can accommodate a
great deal of different terrains unlike triangulation which
requires the survey to be performed along a rigid polygon
shape.
 Only a few observations need to be taken at each station,
whereas in other survey networks a great deal of angular
and linear observations need to be made and considered
 Traverse networks are free of the strength of figure
considerations that happen in triangular systems
7
ADVANTAGES (CONT…)


Trilateration
 Is practical and highly accurate means of rapid control
extension.
 Less expensive than classical triangulation.
 Provides necessary scale control lacking in triangulation.
Satellite positioning






Higher accuracy
Does not require lines of sight between stations
(intervisibility not required)
Economic advantages from greater efficiency and speed of
survey.
Operations are weather independent.
Network independent site selection, hence sites placed
where needed.
3-D coordinates are obtained
8
DISADVANTAGES

Triangulation


Precise traversing



All distances need to be measured.
Lacks the automatic checks on the angle observations as
compared to triangulation.
Trilateration


Intervisibility between stations/points is a prerequisite
or must.
Has a smaller number of internal checks compared to
classical triangulation
Satellite positioning


High capital cost of GPS instrumentation
No sky obstructions can be tolerated, therefore cannot
be used underground, under foliage or structures
9
INTRODUCTION TO TRIANGULATION





“Triangulation consists of a series of connected triangles
which adjoin or overlap each other, angles being measured
from determined fixed stations. Triangulation reduces the
number of measures that need to be taped and for this reason
is often a preferred method of survey. A known base-line
measurement is required.”
TRIANGULATION – A method of surveying in which the
stations are points on the ground at the vertices of a chain or
network of triangles. The angles of the triangles are measured
instrumentally and the sides are derived by computation from
selected sides or bases, whose lengths are obtained by direct
measurement on the ground or by computation from other
triangles.
Triangulation is a surveying technique in which unknown distances
between stations may be determined by trigonometric applications of
a triangle or triangles.
In triangulation, one side is called the baseline and at least two
interior angles of the triangle must be measured.
If all 3 interior angles are measured, accuracy of the calculated
distances is increased and a check provided against any measurement
error.
10
TRIANGULATION NETWORK
11
PRINCIPLE OF TRIANGULATION
12
GEODETIC TRIANGULATION (CONT…)
The recommended length of a base line is usually one sixth to one
fourth of that of the sides of the principal triangles.
TRIANGULATION STATION – A marked point on the earth whose
position is determined by triangulation.
13
GEODETIC TRIANGULATION

Stations will be far apart.

Need to consider the shape and size of the earth. WHY????



Involve a series of triangles called “triangulation system or
network” to complete the control of a selected area
Control is carried from one known base line through
several triangles before another base line is/must be
established or checked into.
Tighter control is obtained by using a series of
quadrilaterals, requiring three other stations to be visible
from each station instead of the two necessary when using
triangles.
14
TYPES OF TRIANGULATION

Two principal types: continuous and chain

CONTINUOUS TRIANGULATION
Advantages:



More accurate
Easier for topographers
CHAIN TRIANGULATION
Advantages:





Easier to adjust before 1950
Less costly
Observed quickly
Chains are the basis of most world mapping, but are now obsolete.
They form grid-iron frame often running north-south and east-west
and about 160-250km apart.
15
CLASSIFICATION OF TRIANGULATION SYSTEM

Primary triangulation (First order)



Average length of sides of triangles is 50km
Angles less than 40 degrees must be avoided (to ensure wellconditioned triangles).
Secondary triangulation (Second order)
Consist of simple homogeneous network of well conditioned
triangles with sided averaging 12km and connected to primary
stations
 There should rarely be more than 10 secondary stations and 15
secondary triangles within a primary triangle.
 Angle greater than 150 and less than 30 degrees should be
avoided.


Tertiary triangulation (Third order)
 Sides average 5-8km and generally points are fixed by
countersection, resection or intersection.
 Very often tertiary points include main and intermediate points.
16
CLASSIFICATION OF TRIANGULATION
SYSTEM (CONT…)





Primary (First-order) triangulation is used to determine the
shape and size of the earth or to cover a vast area like a whole
country with control points to which secondary triangulation
system can be connected.
Second-order triangulation system consist of a network within
a first-order triangulation.
It is used to cover areas of the order of a region, small country or
province
Third-order triangulation is a framework fixed within and
connected to a second-order triangulation system.
It serves the purpose of furnishing the immediate control for
detailed engineering and location surveys.
17
CLASSIFICATION OF TRIANGULATION
SYSTEM (CONT…)
18
TRIANGULATION FIGURES AND LAYOUTS

The basic figures used in triangulation networks are the
triangle, braced or geodetic quadrilateral and the polygon
with a central station.
19
TRIANGULATION LAYOUTS

1.
2.
3.
4.
5.
The triangles in a triangulation system can be arranged in
a number of ways:
Single chain of triangles
Double chain of triangles
Braced quadrilaterals
Centered triangles and polygons
A combination of the above
20
1. SINGLE CHAIN OF TRIANGLES


Is rapid and economic due to its simplicity of sighting only
four other station and does not involve observations of long
diagonals.
Simple triangles of a triangulation system provide only one
route through which distances can be computed and hence
does not provide any checks on the accuracy of
observations.
21
2. DOUBLE CHAIN OF TRIANGLES


This arrangement is used for covering the larger width of a
belt.
The system has disadvantages of single chain of triangles
system.
22
3. BRACED QUADRILATERALS




A triangulation system consisting of figures containing four
corner stations and observed diagonals.
The triangles are overlapping.
This system is treated to be the strongest and the best
arrangement of triangles, and it provides a means of
computing the lengths of the sides using different
combinations of sides and angles.
Most of the triangulation systems use this arrangement.
23
4. CENTERED TRIANGLES AND POLYGONS




A triangulation system which consists of figures containing
interior stations in triangle and polygon.
Is generally used when vast area in all directions is required to be
covered.
The centered figures generally are quadrilaterals, pentagons, or
hexagons with central stations.
Though this system provides checks on the accuracy of the work,
generally it is not as strong as the braced quadrilateral
arrangement.
24
LAYOUT OF PRIMARY TRIANGULATION

Two types of frameworks of primary triangulation are
provided for a large country to cover the entire area:
1.
2.
Grid iron system
Central system
25
1. GRID IRON SYSTEM



In this system, the primary triangulation
is laid in series of chains of triangles,
which usually run roughly along the
meridians (north-south) and along
perpendiculars to the meridians (eastwest), throughout the country.
The distance between two such chains
may vary from 150-250km.
The area(s) between the parallel and
perpendicular
series
of
primary
triangulation are filled by the secondary
and tertiary triangulation systems.
26
2. CENTRAL SYSTEM


In this system, the whole area is covered by a network of primary
triangulation extending in all directions from the initial
triangulation figure ABC, which is generally laid at the centre of
the country.
This system is generally used for the survey of an area of
moderate extent.
27
CRITERIA FOR SELECTION OF THE LAYOUT OF
TRIANGLES

1.
2.
3.
4.
5.
The under mentioned points should be considered while
deciding and selecting a suitable layout of triangles.
Simple triangles should be preferably equilateral.
Braced quadrilaterals should be preferably
approximate squares.
Centered polygons should be regular.
The arrangement should be such that the
computations can be done through two or more
independent routes.
The arrangement should be such that at least one
route and preferably two routes form well
conditioned triangles.
28
CRITERIA FOR SELECTION OF THE
LAYOUT OF TRIANGLES
(CONT…)
6. No angle of the figure, opposite a known side should be
small, whichever end of the series is used for
computation.
7. Angles of simple triangles should not be less than 45°, and in
the case of quadrilaterals, no angle should be less than
30°. In the case of centered polygons, no angle should be
less than 40°.
8. The sides of the figures should be of comparable lengths.
Very long lines and very short lines should be avoided.
9. The layout should be such that it requires least work to
achieve maximum progress.
10. As far as possible, complex figures should not involve more
than 12 conditions.
It may be noted that if a very small angle of a triangle does not
fall opposite the known side it does not affect the accuracy of
triangulation.
29
WELL CONDITIONED TRIANGLES


The accuracy of a triangulation system is greatly affected
by the arrangement of triangles in the layout and the
magnitude of the angles in individual triangles.
The triangles of such a shape, in which any error in
angular measurement has a minimum effect upon the
computed lengths, is known as well-conditioned triangle.
30
STRENGTH OF FIGURE


The strength of figure is a factor to be considered in
establishing a triangulation system to maintain the
computations within a desired degree of precision.
It plays also an important role in deciding the layout of a
triangulation system.
31
SUMMARY










Definition of geodetic network
Purpose of geodetic network
Methods of formation
Advantages and disadvantages
Triangulation principle
Types of triangulation
Classification of triangulation
Triangulation figures and layouts
Selection criteria of layouts
Well conditioned triangles and strength of a figure
32
QUESTIONS????
GEODETIC TRIANGULATION
OF ZIMBABWE
OVERVIEW

History of Triangulation of Zimbabwe

Routine of triangulation survey

Reconnaissance

Intervisibility
35
HISTORY OF TRIANGULATION OF
ZIMBABWE

Simms Chain (1897-1901)
Extended by Alexander Simms from Bulawayo to Harare
and onto the Zambezi covering 4o of the arc of the 30th
meridian i.e. 30o latitude.
 Bases were measured at Insiza and Gwebi and longitude
and latitude observed at Bulawayo and Harare.


Rubin (1903-1906)


Extended the arc of the 30th meridian northwards over
another 7o within 75 miles of Lake Tanzania.
Gordon’s Chain (1906)

The gap between the Simm’s system and the Transvaal
system was bridged by Captain Gordon.
36
37
ESTABLISHMENT OF TRIGONOMETRICAL
SECTION



A period of 22 years elapsed before the trig section of the
SG department formed in 1929.
This was inspired by the need for trig controls in the copper
queen mine area which highlighted a need for continuation
of Geodetic and primary control in the country as a whole.
Concrete pillars (4 feet by 19 inches) were introduced for
the first time, trig surveys began using Wild T3.
38
COMPLETION OF SERIES AND CIRCUITS
(1930-1939)










In 1930, the so called Umtali series was started consisting of a double
chain of triangles or polygons.
At the Mozambique boarder, the series turned south and continued
down as far as 20o South.
The Victoria series connected Simm’s system and the Umtali series at
20o south thus completing circuit 7.
In 1936, the Nuanetsi series was started at the Victoria series and run
to make a second connection to the Transvaal system.
Circuit II was now complete, the Sabi base was measured at the
junction of Victoria and Umtali series by Major Hotine.
In 1938, Simm’s chain was widened and strengthened to form the
Simm’s series.
The new Gwebi and new Insiza bases were also measured.
By 1939, all fieldwork and observations were complete for the 2 main
Geodetic circuits of Zimbabwe.
Trig survey was then closed for the duration of the 2nd World war.
In 1946, northern Transvaal triangulation data was made available
and the circuit adjustments were completed thus making the
Zimbabwe and South Africa homogeneous for the 1st time.
39
RECOMPUTATION OF THE AFRICAN AREA


In 1946, Simms triangulation North of circuit and the
Simms Rubin connection was strengthened.
This cleared the way for the recomputation of the arc from
Zimbabwe to Uganda by the directorate of colonial surveys
in December 1951.
40
EXTENSION BY PRIMARY NETWORKS AND
SERIES




The initial framework was gradually filled in and expanded
by primary triangulation in the form of networks and
series.
The last major primary task covered the north east section
of the country (Gonarezhou) in 1972-1973.
There has been also a steady breakdown to secondary,
tertiary, etc and also precise leveling and TSM work.
In 1953, the 4th and last Geodetic base was introduced in
1956 and several hundred primary and secondary lines
have been measured with a variety of Tellurometer.
41
ROUTINE OF TRIANGULATION
SURVEY


i.
ii.
iii.
iv.
v.
vi.
The routine of triangulation survey, broadly consists of
(a) field work, and (b) computations.
The field work of triangulation is divided into the following
operations :
Reconnaissance
Erection 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.
42
RECONNAISSANCE



Reconnaissance is the preliminary field inspection of the entire area to
be covered by triangulation, and collection of relevant data.
Since the basic principle of survey is working from whole to the part,
reconnaissance is very important in all types of surveys. It requires
great skill, experience and judgment.
The accuracy and economy of triangulation greatly depends upon
proper reconnaissance survey. It includes the following operations:
1. Examination of terrain to be surveyed.
2. Selection of suitable sites for measurement of base lines.
3. Selection of suitable positions for triangulation stations.
4. Determination of intervisibility of triangulation stations.
5. Selection of conspicuous well-defined natural points to be used
as intersected points.
6. Collection of miscellaneous information regarding:
(a) Access to various triangulation stations
(b) Transport facilities
(c) Availability of food, water, etc.
(d) Availability of labour
(e) Camping ground.
43
RECONNAISSANCE (CONT…)








Reconnaissance may be effectively carried out if accurate
topographical maps of the area are available.
Help of aerial photographs and mosaics, if available, is also
taken.
If maps and aerial photographs are not available, a rapid
preliminary reconnaissance is undertaken to ascertain the
general location of possible schemes of triangulation suitable
for the topography.
Later on, main reconnaissance is done to examine these
schemes.
The main reconnaissance is a very rough triangulation.
The plotting of the rough triangulation may be done by
protracting the angles.
The essential features of the topography are also sketched in.
The final scheme is selected by studying the relative strengths
and cost to various schemes.
44
GROUND RECONNAISSANCE

This includes the following:
i.
Testing and determination of the visibility of all rays.
ii.
Final selection and marking of stations.
iii. Specifying of type and dimensions of beacons
iv. Observing of sufficient directions, vertical angles to
plot the scheme and enable stations to be reidentified.
v.
Recording of means of access and availability of
beacon material.
vi. *NB During reconnaissance sights are cleared along
proposed lines and flags are erected.
vii. Reconnaissance ladders can be essential in the case of
sights for elevated beacons.
viii. Lamps can be used to test lines of sight in poor
visibility.
ix. Sites are marked by a peg or stone in the ground
covered by a stone cairn, a paint mark on the rock site
with cairn.
45
RECONNAISSANCE (CONT…)

For reconnaissance the following instruments are generally
employed:
1.
2.
3.
4.
5.
6.
7.
8.
Small theodolite and sextant for measurement of
angles.
Prismatic compass for measurement of bearings.
Steel tape.
Aneroid barometer for ascertaining elevations.
Heliotropes for ascertaining intervisibility.
Binocular.
Drawing instruments and material.
Guyed ladders, creepers, ropes, etc., for climbing trees.
46
DETERMINATION OF INTERVISIBILITY OF
TRIANGULATION STATIONS

The rule for intervisibility between proposed status A, B is
defined by the equation
Where
h – height of point on light ray
h1 – height of A
h2 – height of B
d1 – distance in km of point from A
d2 – distance in km of point from B
47
EXAMPLE


Determine the intervisibility between 2 points C and D, hC
= 1560, hD = 1780 for a point whose distance from C and D
are 4507 and 6750m respectively.
If the point is called P and hp = 1450m. Calculate if P is
intervisible.
H = 1560 +(1780-1560) (
) - 0.067(4.507)(6.750)
= 1646m
Since calculated is more than Hp (1450), the two points will be
intervisible.
48
SUMMARY

History of Triangulation of Zimbabwe

Routine of triangulation survey

Reconnaissance

Intervisibility
49
QUESTIONS????
SURVEYING III
TRIGONOMETRICAL BEACONS
Juliana Useya
Department of Geoinformatics and Surveying
OVERVIEW OF TODAY’S LECTURE

Beacon establishment and maintenance




Construction
Air marks
Inspection
Witness marks

Triangle closure and spherical excess

Observations at trig stations







Observing techniques
Elimination of micrometer, circle graduation errors
Laplace station
Base extension
Horizontal fixing techniques
Triangulation tolerances
Witness mark consistency check
52
TRIGONOMETRICAL BEACON
53
54
4 FOOT PILLAR WITH 6 INCH PEDESTAL
BUILT ON SOLID ROCK
55
BEACON CONSTRUCTION

The standard beacon is a 4 feet by 18 inches complex
cylindrical pillar on a 6 inches pedestal on which name
and number are inscribed.







This includes three to five half inches reinforcing iron rods
from base to top of pillar and a drain pipe channeling water
away from a centre pipe.
This reduces corrosion
Lightning conductors (copper) are also fitted on all beacons
built on solid rock
Some stations consist of pedestal up to 10 m in height to
which ladders and guide rails are fitted.
There are 2 standard signal sizes: Primary and secondary,
tertiary and lower order stations
After primary observations have been completed, the
primary veins are removed and replaced by secondary veins.
The primary signals are large enough to be comfortably
bisected in good visibility at 70km.
56
AIR MARKS

These are normally constructed of white washed stones
forming either Y or circular shapes for aerial photography.
57
INSPECTION



Because of the cost of establishment of trig stations in a
country, It is necessary to have a program of regular
inspection and repair.
This enables title surveys to be connected to the trig
system as well.
The policy of the Surveyor General is to inspect each trig
station once every 5 years, clearing secondary vegetation
growth and trees, repainting beacons and signals and
repairing damaged or destroyed points.
58
WITNESS MARKS




Each trig station is related to 3 or 4 secret marks
established at suitable points near to and symmetrically
spaced around the beacon.
The purpose of these, is to define and safeguard the exact
station of the centre mark in case of movement by
whatever means.
The marks are either iron pegs in concrete below ground
level with cairn or pegs set in grouted holes and placed
about 7m away at a spacing of roughly 120o.
The tops of the pegs should be visible from the pillar and
the distance between them should be easy to measure.
59
OBSERVATIONS
The observations at a trig station generally consist of the
following:
1.
2.
3.
4.
Vertical angles
Horizontal angles
Witness marks and farm beacon observations
Minor point observation etc.
60
OBSERVING TECHNIQUES FOR ALL ORDERS
OF HORIZONTAL AND VERTICAL ANGLES
I.
II.
III.
IV.






Theodolite must be centred and leveled at least 10 minutes before the
observations begin.
Shading by umbrella is essential
Just prior to observing, the instrument must be checked for level.
During horizontal angle measurement, the vertical circle should be
left unclamped.
Final vertical bisection can be achieved by gently tapping the telescope
Observations must be made to well defined targets.
When in any doubt about bisecting an opaque signal, replace with
luminous signal e.g. lamp or place a screen behind
Do not bisect pillars because of haze
Vertical angles are measured to top pillar, top or bottom of vain, or
centre of lamp and the index bubble must always be leveled prior to
observations.
Vertical angles must be observed between 1100hrs (11am) and 1500hrs
(3pm). WHY??????????
61
ELIMINATION OF MICROMETER, CIRCLE
GRADUATION ERRORS

This can be minimised by using different zeros thereby
spreading the error over the whole circle.
Specifications
 Primary – 16 arcs with a Wild T3
 Secondary – 8 arcs with a Wild T3
 Tertiary – 4 arcs with a Wild T3 or T2
 Quaternary – 4 arcs with a Wild T3 or T2
62
SUMMARY OF TRIANGULATION TOLERANCES
63
RECORDING OBSERVATIONS




Horizontal and vertical angles are read and recorded with a
precision required by the order of triangulation.
Prior to observing, sheets should be prepared and setting
data filled in, include notes on visibility, background and
signal so as to weigh your observations.
The RO misclosure on each face and mean of the 2 must
not exceed the specific amount of the order of triangulation.
The range (time) over the arcs observed must not exceed
the maximum allowed for that order.
64
ELIMINATION OF PROJECTION
Error in Triangulation

Owing to the possibility of the horizontal circle not being
orthogonal to the vertical axis, the theodolite should be
turned bodily 90o in the middle of observation.
65
TRIANGLE CLOSURE AND SPHERICAL
EXCESS

In the case of primary and secondary triangles, spherical
excess needs to be calculated and taken into consideration
before determining misclosure amount.
∆ - Area of triangle
Pm – mean radius in prime meridian
⋎m - mean radius in prime vertical
R - mean radius
R ≠ Pm ≈ ⋎m
It is the amount by which the sum of angles of a spherical triangle
exceeds 180o.
Triangular misclosure = (A+B+C) – (180o+ E’’)
66
EXAMPLE: TRIANGULAR MISCLOSURE


In a geodetic survey, the mean angles in table below were
observed in one triangle, each having been observed the same
number of times under similar conditions.
Station
Mean value
A
62⁰ 24’ 18.4’’
B
64⁰ 56’ 09.9’’
C
52⁰ 39’ 34.4’’
Side AB was known to be 37 269.280 m long. Estimate the
corrected values of the three angles. Take the radius of the
earth to be 6383.393km.
67
SOLUTION

Calculate spherical excess
 Assume triangle is plane, calculate sum of the 3 mean angles
A+B+C =180° 00’ 02.7’‘
Hence deduct 2.7’’/3 = 0.9’’ from each angle.
 Use sine rule to calculate one other side (BC = 41 544.469m)
 Area of a triangle (1/2 a.b Sin C = 701.3 km²)
 Spherical excess
= 3.6’’


Theoretical sum of the angles = 180 + spherical excess = 180° 00’ 03,6’’

Sum of observed angles = 180° 00’ 02.7’‘

Triangular misclosure = ∑observed angles – (180+spherical excess )= -0.9’’
Adjusted angles
A
62⁰ 24’ 18.7’’
B
64⁰ 56’ 10.2’’
C
52⁰ 39’ 34.7’’
68
OBSERVATIONS AT A TRIG STATION
1. At a trig station:
 An arc of the face left and face right of horizontal
observations to include all the witness marks, farm
beacons, surveyors stations and boundary pillars in the
immediate vicinity and to distinct trig stations.
 Measure slope distance and vertical angles to each of the
witness marks and other nearby points.
 Measure the height of the horizontal axis above top of the
pillar or station marks to enable determination of reduced
levels.
2. At each witness mark
 Write detailed description of each mark e.g. 2mm iron peg
in concrete, WM1 inscribed in cement and grouted in rock.
 Measure horizontal distance between witness marks to
provide a computational check.
 *N.B. all distances are measured with a steel tape and
recorded to 0.005m.
69
OBSERVATION OF WITNESS MARKS
3. The consistency check
 Computation must agree within ±0.010m.
 When there is no consistency measurement taken, the
relevant line is not drawn in on the Witness Mark sheet or
WM sketch.
4. Beacon and site report (verification and comparison
of Witness Marks).
 Witness marks’ observations are observed and recorded at
all trig stations visited in the course of the observing
program whether they are new or established stations.
 If at the established station, a movement is experienced or
disagreement on certain measurements detected,
observations are entered on a supplementary beacon, site
and access report forms.
 Signal verticality is tested by use of a plumb bob.
 If an appreciable lean is experienced, a test must be made
instrumentarily.
70
LAPLACE STATIONS


These are points in a network at which geodetic azimuth is
determined by observing Astro-Azimuth and longitude.
Geodetic Azimuth = Astro-Azimuth – (Astro-Longitude – Geolongitude)sin ф
Where:
∝ - Geodetic Azimuth
∩ - east-west deviation of the vertical
A – Astronomical azimuth
λ - geodetic longitude
Λ – Astronomical longitude
Ф – geodetic latitude
71
BASELINES




For invar tape baselines were measured in Zimbabwe,
Hwange, Gwebi, Sabi and Insiza, these average 12km.
Measurements were carried out in both directions by 2
wires used simultaneously (four in all).
The tapes or wires were standardized before and after the
base measurement and wind screens (buffer) were used to
shield the tape were necessary.
In addition, the tripod heads were spirit leveled.
72
BASELINES (CONT…)
Corrections to the taped
readings

Sag, temperature, slope,
deformation due to difference in
heights of tape supports tension
i.e. change in elastic,
elongation, reduction to sea
level i.e. MSL correction,
standardization.
Baseline extension


The initial size of the geodetic
chain could not be measured
directly , until the advent of
long range EDM.
Instead baseline extension was
the method employed.
73
BASELINES (CONT…)




Referring to the diagram, AB is the correct baseline of say
10km.
If the angles marked in triangles ABCA and ABDA are
observed with a theodolite, then the length of line CD may
be calculated.
This line CD, now becomes the new extended baseline of
say 25km in length.
By taking more angle observations on to 1/P and 2/P from
C and D, this new baseline can be extended to give the
length of 1/P to 2/P, which is the initial side of the geodetic
chain of about 60km.
74
OTHER HORIZONTAL FIXING TECHNIQUES
1.
2.
EDM traversing
Trilateration
1. EDM traversing








Involves fixing of points by measuring distances and
directions from control points.
Advantage over triangulation
Ease of design (points close to each other)
Easy of organizing observations (use few arcs).
No accumulation of scale error.
Communication via radio
Ease of adjustment compared to chains or nets.
Lengths of traverse legs are typically 10-20km flat country
using towers are around 40km in hilly areas.
Laplace stations should preferably be located at every
alternate station or at least once at every 6th station.
75
OTHER HORIZONTAL FIXING TECHNIQUES
(CONT…)
2. Trilateration

A continuous net of laser geodimeter lines with satellite or
laplace azimuth control and adjusted as a whole is probably
the most accurate of terrestrial networks.
76
EXAMPLE: REDUCTION OF T3 THEODOLITE
OBSERVATIONS
77
78
EXAMPLE 2
PERFORM A WITNESS MARK CONSISTENCY CHECK FOR THE
FOLLOWING OBSERVATIONS TAKEN WITH A T2 THEODOLITE.
ARE THE OBSERVATIONS CONSISTENT?
79
ADDITIONAL INFORMATION
Horizontal distances
 WM1 – WM2 – 19.654
 WM2 – WM3 - 20.560
 WM3 – WM1 - 16.594
80
ANSWER
JOINS
T9/Q – GPS1
05 24 24
T9/Q – PV/Q
132 09 42
MEAN ORENTATION = (2+0)/2
=1
POLARS
@T9/Q
Horizontal distance
WM3
84 08 58
9.508
WM1
84 48 42
7.658
WM2
87 06 00
16.095
81
ANSWER
82
ANSWER


Limit = ±0.010m
There is consistency between the previously observed
witness marks. So we assume the beacon hasn’t moved.
83
SUMMARY

Beacon establishment of and maintenance




Construction
Airmarks
Inspection
Witness marks

Triangle closure and spherical excess

Observations at trig stations







Observing techniques
Elimination of micrometer, circle graduation errors
Laplace station
Base extension
Horizontal fixing techniques
Triangulation tolerances
Witness mark consistency check
84
QUESTIONS????
SURVEYING III
RESECTION AND
INTERSECTION
Juliana Useya
Department of Geoinformatics and Surveying
OVERVIEW OF TODAY’S LECTURE

RECAP:

Resection

Intersection
87
RESECTION





This is a method of determining the coordinates of an
unknown point by observing at least 3 controls.
The observations are taken from the unknown point.
There are no orientation forward rays from the known
points to the unknown point, which is the main
disadvantage of resection.
The point to be resected can either be within or outside the
triangle defined by the 3 trig station.
When it is within, the solution is normally perfect but when
outside a DANGER CIRCLE might result.
88
DANGER CIRCLE


Occurs when the 3 points being used in resection and the
unknown points fall on the circumference subtended by the
same chord.
Resection is employed when the trigs are far moved from
the area under survey, such that trying to occupy them will
take too much time.
89
NATURE OF RESECTION DIAGRAMS


POINTS INSIDE (MOST IDEAL)
Lettering (ABC) should be clockwise
90
POINT OUTSIDE
91
BARYCETRIC METHOD FOR SOLVING RESECTION
PROBLEMS
where
92
CHECK
(YP-YA)K1 + (YP-YB) + (YP-YC)K3 = 0
 (XP-XA)K1 + (XP-XB) + (XP-XC)K3 = 0

93
EXAMPLE : RESECTION


@P
T1 00 00 00
T2 142 43 12
T3 196 56 28
If coordinates of T1, T2 and T3 are
Y
X
T1
6 205.04
76 699.95
T2
5 898.88
72 661.79
T3
4 939.24
75 445.54
94
EXAMPLE: RESECTION (CONT…)

Calculate coordinate of P
95
ANSWER: RESECTION (CONT…)
∝ = (P T3 – PT2) = 54 13 16
𝛃 = (P T1 – PT3) = 163 03 32
⋎ = (P T2 – PT1) = 142 43 12
1.
2.
Check: ∝ + 𝛃 +⋎= 360⁰
JOIN DIRECTION: T1 –T2
96
INTERSECTION



Is a process of locating and coordinating a point from at
least 2 existing control points/stations by observing
horizontal directions to the point to be fixed.
The point to be fixed is not occupied.
Used in surveying details, in accessible places e.g. flag pole,
church spires.
97
CALCULATING COORDINATES FOR
INTERSECTION
Coordinates can be obtained by use of angles and/or
directions.
1. using angles

98
CALCULATING COORDINATES FOR
INTERSECTION (CONT…)



USING DIRECTIONS
PL = AL tan ∝ = YP – YA = (XP -XA) tan ∝ ------------- 1
PM = BM tan 𝛃 = YP – YB = (YP - YA) tan 𝛃 --------------2
99
INTERSECTION



Subtracting 2 from 1
YB – YA = (XP - XA)tan ∝ - ( XP - XB)tan 𝛽
= Xp – tan 𝛼 –XA tan 𝛼 - XP tan𝛽 + XB tan 𝛽
= XP (tan𝛼 – tan 𝛽) + XB (tan 𝛽 – XA tan 𝛼)
Using equation 1
∆YAP = ∆ XAP tan ∝
YP = YA + ∆YAP
IT IS ALSO POSSIBLE TO USE THE FOLLOWING FORMULAE:
∆YAP = ∆XAP tan 𝛼
where
𝛼 – dir AP
𝛽 – dir BP
100
EXAMPLE ON INTERSECTION


Calculate coordinates of point K which was intersected
from 101/P and 382/T using the following information.
Coord
Y (m)
X (m)
101/P
-4 065.67
+ 73 032.29
382/T
-4 829.26
+ 71 535.13
Direction observation
101/P to K 171⁰ 47’ 56’’
382/T to K 84⁰ 04’ 56’’
∝ = 171⁰ 47’ 56’’
𝛃 = 84⁰ 04’ 56’’
101
QUESTIONS????
SURVEYING III
ECCENTRIC STATION AND
3RD
ORDER TRIANGULATION
Juliana Useya
Department of Geoinformatics and Surveying
OVERVIEW

Eccentric station

Equal shifts adjustments

3rd order triangulation

Cross cuts

Error figure
104
ECCENTRIC STATION




Observations of certain rays may not be possible from the
trig station owing either to the nature of the site or to the
features on the line of sight.
In such a situation, eccentric stations can be used to
overcome the problem.
However, after completion of the observations, the
directions have to be reduced such that they appear as
being observed from the trig station.
This is done by applying some corrections to measured
directions.
105
ECCENTRIC STATION (CONT…)



Point R cannot be occupied in triangle RST and at satellite station X,
the directions to R,S&T are observed and the horizontal distance d
between the trig station R and eccentric station X is measured.
Angles ∝1 and ∝2 are derived from the observation.
The direction observed at station X can be reduced to their
equivalence at R by applying corrections C1 and C2 obtained from
where
d – distance from R to eccentric station X
we assume that distance RT ≈ XT


Direction R - S = dir XS + C1
Direction R - T = dir XT - C2
106
ADJUSTMENT OF TRIANGULATION
OBSERVATIONS




Adjustment of Triangulation observations (horizontal)
depending on the order of triangulation, angle observations
in triangulation schemes have to be adjusted for geometric
consistency before any coordinate calculations are
attempted.
In case of 1st and 2nd order triangulation, the method of
equal shifts is normally adopted whilst for the third order a
direction sheet can be extracted and adjustments
performed.
For simple triangles, equal shift adjustment is achieved by
taking each triangle in turn through the network,
calculating the misclosure and applying 1/3 of this to each
angle.
Methods of adjusting the observed angles in braced quads
and center point polygons by equal shift are given below.
107
ADJUSTMENT OF A BRACED QUAD USING
THE EQUAL SHIFTS METHODS
108
PERFORMING THE ANGLE ADJUSTMENT

3 angle conditions has to be satisfied for the equal shifts
adjustment, these are:
i.
ii.
iii.

∑angles (1-8) = 360o
Angle 1+2 = Angle 5+6
Angle 3+4 = Angle 7+8
Conditions ii and iii are known as adjustment to opposites
109
PERFORMING THE SIDE ADJUSTMENT

In addition to angle conditions, a side condition must also be satisfied for a
braced quad i.e.

This non-linear condition is applied to the first adjusted angles by calculating
the side adjustment (V) from
where
a = Sin 1 Sin 3 Sin 5 Sin 7
b = Cot 1 + Cot 3 + Cot 5 + Cot 7
c = Sin 2 Sin 4 Sin 6 Sin 8
d= Cot 2 + Cot 4 + Cot 6 + Cot 8

The value of |V’’| is computed using the first adjusted angles and if c > a, the
value of |V’’| is subtracted from each of the even angles and added to each of
the odd angles.
110
Angle
Observed
value
Adjustm
ent to
all
angles
Adjustme
nt to
opposites
First
adjusted
angle
Side
adjust
ment
Final
adjusted
Final
Angle
1
50 21 19
+1.12’’
-2’’
50 21 18.12
+5.01
50 21 23.13
50 21 23
2
56 21 41
+1.12’’
-2’’
56. 21 40.12
-5.01
56 21 35.11
56 21 35
3
43 19 39
+1.12’’
-1.75’’
43 19 38.37
+5.01
43 19 43.38
43 19 43
4
29 57 24
+1.12’’
-1.75’’
29 57 23.37
-5.01
29 57 18.36
29 57 18
5
47 21 16
+1.13’’
+2’’
47 21 19.13
+5.01
47 21 24.14
47 21 24
6
59 21 36
+1.13’’
+2’’
59 21 39.13
-5.01
59 21 34.12
59 21 34
7
34 52 11
+1.13’’
+1.75’’
34 52 13.88
+5.01
34 52 18.87
34 52 19
8
38 24 45
+1.13’’
+1.75’’
38 24 47.88
-5.01
38 24 42.87
38 24111
43
Total
359 59 51
360 00 00
360 00 00
EQUAL SHIFTS ADJUSTMENT FOR A
CENTRED POINT POLYGON

ABC shows a triangulation scheme consisting of points ABC
in the form of a triangulation with a fourth point S in the
centre of the triangle.
112
ANGLE ADJUSTMENT IS PERFORMED IN THE
FOLLOWING MANNER:






When the angle conditions are clearly satisfied in the centre point
polygon, the adjusted angles in a triangle must be 180.
Adjusted angles 7, 8 & 9 must be added to 360 and adjusted accordingly
The side condition for the centre point polygon is similar to that of a
braced quad, in this case
And the side adjustment is obtained used an expression identical to the
braced quad
However, only the outer angles are used to give
a = Sin 1 Sin3 Sin 5
b = Cot 1 + Cot 3 + Cot 5
c = Sin 2 Sin 4 Sin 6
d = Cot 2 + Cot 4 + Cot 6
NB. The value of v is calculated using the angles obtained after applying
113
the angle conditions.
3RD ORDER TRIANGULATION
COUNTERSECTION; REDUCTION OF COUNTERSECTION
OBSERVATIONS



1.
2.
3.
4.
5.
Observations taken by combining resection and intersection to
fix a single point result in excess observations being taken.
To reduce the observations a number of steps have to be taken.
Calculation procedures involve the following:
Reduction of angle observations
Entering observations in a direction sheet and reducing them
Calculation of provisional coordinates (This is normally done
using the intersection method)
Calculation of cross cuts (done for the remaining rays i.e.
using points not used in calculation of provisional
coordinates)
Plotting of an error figure and solving
NB each ray must be assigned a weight which is proportional to
its distance i.e. w ∝ 1/s2
114
3RD ORDER TRIANGULATION (CONT…)
1. Reduction of angle observations
 Triangulation observations should always be made on both
faces of the theodolite.
 Reductions are therefore conducted by finding the mean of the
observations
2. Abstract of observations
 At all stations, the surveyor will probably observe more than
one arc, hence in order to adjust the directions, an abstract of
observations has to be extracted.
 Here RO correction is applied
3. Direction sheet
 Adjusted mean directions are entered in a direction sheet.
 This operation is required in order that the observed direction
may be oriented as nearly as possible to their true values for
use in the individual calculation
115
3RD ORDER TRIANGULATION (CONT…)
4. Calculation of provisional Coordinates
 This is done using intersection method
 2 controls which provide the strongest fix are selected
 It follows that the points to be used should have angles
falling between 30o and 120o as subtended at the unknown
point.
 Points or controls forming an angle of 90o will be the best.
116
3RD ORDER TRIANGULATION (CONT…)
5. Calculation of cross-cuts
 The intersection calculation is used to determine the
provisional values of a point to be calculated by consideration
of only 2 observed rays.
 To fix the position of any point by triangulation, at least 3
observations are required from known positions, 2 of these
rays are used in the intersection, so what of the 3rd or other
rays observed?
 These rays are now used to find their relative positions in
relation to the provisional values to form triangles of error
 Cross cuts involve calculation where the remaining rays cut
the x and y grid which passes through the provisional values.
 y=y1
 dx is greater when the direction is generally in a north south

The calculation on whether the ray will cut the y or x grid will
depend on the direction of the ray.
117
CROSS CUTS
118
CROSS CUTS (CONT…)

NB in the case of dy greater, tan ∝ = dy/dx, whereby we are
using directions not angles so

** If direction is closer or nearer to the y axis, dx=dy cot∝.

If direction is nearer to the x axis dy = dx tan ∝

Directions for the cross cuts are obtained from the oriented
back directions.
119
SUMMARY FOR 3RD ORDER
TRIANGULATION CALCULATION
120
EXAMPLE:
Using the following information obtained from a countersection
scheme of triangulation.
i.
ii.
iii.
iv.
Reduce the direction by applying all necessary corrections and
extract a direction sheet.
Calculate the cross cuts for the remaining rays.
Compute weights for all rays necessary in the drawing of the
error figure
Solve the problem using an error figure
121
COORDINATE LIST GAUSS LO 31⁰

N.B. (t-T) correction is 0 for all observed rays
122
FIELD BOOK
123
124



Provisional coordinates of P
Y
(A) 101/P
-4065.67
(B) 382/T
-4829.26
X
+73032.29
+71535.13
(A)- 101/P-P = ∝ = 171 47 56
(B)- 382/T-P = 𝛃 = 84 04 56
125
∆YAP = ∆XAP tan∝
∆YAP = -763,59
∆XAP = -1497.16
Tan ∝= -0.144121996
Tan 𝛃 = 9.647522764
Tan ∝ – tan 𝛃 = -0.144121996 – 9.647522764 = -9.79164476
∆XAP = -1397.14
∆YAP = 201,36
Provisional coordinates of P
Y
P
-3864.31
Distance 101/P – P = 1412m
X
+71 635.15
For checking USE 382/T as reference point
+964.95
+100.02
Distance 382/T – P = 970m
126
CROSS CUTS
For 137/T
137/T
P
∝ = 215 44 28 (Closer to x-axis ∆Y = ∆X tan ∝)
-1326.15
+75161.98
-2538.12
+71635.15
-3864.27
-3526.83
-3864.31
+0.04
Calculate ∆Y = ∆X tan ∝
∆Y = -3526.83 tan 215 44 28
= -2538.12
Y coordinate for P = -1326.15 + (-2538.12) = -3864.27 – 3864.31 = 0.04
Distance of SP – 137/T = 4345m
For 82/Q ∝ = 80o 22’ 52’’ (Close to Y axis, ∆Y = ∆Xtan∝, ∆X=∆Ytan∝)
∆X = ∆Ycot ∝ = 256.11
82/Q
-5375.49
+71379.07
P
-3864.31
+ 71635.18
∆Y
+1511.18
+ 256.11
+71635.18
+71535.15
0.03
Distance = 1533m
127
WEIGHTS

We are multiplying the observations from382/T-101/P and
101/P-382/T by 2 bcz, it will give us a double weight as it is
more accurate.
128
ERROR FIGURE




With several possible values for the coordinates of P, it is
clear that an adjustment is necessary.
This time the adjustment is best done by graphical means.
Thus there is enough data from the triangle computation
and from the cross cut computations to draw graphical
representation of the actual position of the fixing rays with
relation to each other.
The resultant figure will have several solutions to obtain
the most probable position for the point P.
129
ERROR FIGURE (CONT…)


The plotting of the error figure is fairly simple operation.
To start off with, a convenient position is selected to depict
the position of the provisional coordinates i.e. coordinates
obtained with rays 382/T and 101/P.

The position of P must be selected at the angular shifts
applied to the rays must be as small as possible.
Cross cuts positions are identified on the appropriate axis
and directions for the particular rays plotted with the use
of a protractor.

Each ray is labeled accordingly.

130
ERROR FIGURE (CONT…)




Since several triangular errors are obtained these
have to be eliminated by combining some other rays
so that a single triangle remains.
Pair of rays which form the smallest angle of
intersection are targeted for the combination.
The resultant ray will be awarded or given a weight
equal to the sum of the weights of the 2 rays.
The weighted centre for the resultant triangle of error
is accepted as the true value of the point i.e. P.
131
QUESTIONS????
SURVEYING III
NATIONAL HEIGHT NETWORK
Juliana Useya
Department of Geoinformatics and Surveying
INTRODUCTION

1.
2.
3.
4.
5.
6.
7.
A knowledge of the heights and levels of points on the earth’s
surface are required for all or some of the following purposes:
To enable survey measurements, aerial photos and satellite
imagery, to be reduced to a datum surface such as sea level or
a reference ellipsoid.
To create DTMs to be used to solve problems such as
intervisibility between points on the ground.
To enable relief to be depicted on topographical and air maps.
To provide information for scientists concerning shape and
structure of the earth.
To enable roads and railways to be constructed in such a
manner that steep hills are avoided.
To enable drainage and other water works to be surveyed so
that water may flow in desired directions.
Etc.
134
HEIGHT DATUMS
The types of height datums are:
1. Geoid – It is that equi-potential surface of the earth gravity field
which best fits the global mean sea level.
At every point, the geoid surface is perpendicular to the local
plumbline.
 Geoidal heights are required for topographic heights above mean sea
level, engineering surveys, hydro-electric projects, gravity stations,
sea-surface topography.

2. Ellipsoidal surface
 Provides ellipsoidal heights required for:
Reduction of measured distances to ellipsoidal distances for
triangulation computation.
 To provide the third item of the triplet (ф, λ, h), for conversion to 3D
cartesian coordinates used with satellites.


The principle height datum is the geoid (mean sea level). Satellite
observations from which height above ellipsoid can be calculated.
135
METHODS OF HEIGHT MEASUREMENT





Spirit leveling (Ordinary or precise)
Trig heighting
Satellite position system (GNSS)
Hydrostatic levelling
Barometric heighting
NB: Spirit leveled heights are very accurate and form the
backbone of the National Height Network,
why?????
because effects of refraction and curvature are minimal.
136
PRECISE LEVELLING





In Zimbabwe, precise spirit leveling is classified as one of
the following:
1. Primary 1st order
2. Secondary 2nd order
3. Tertiary 3rd order
Primary circuits are first chosen and leveled and these are
subdivided into secondary and tertiary routes.
The first primary circuit in Zimbabwe was completed in
1974 with a misclosure of approximately 0.01m over a
distance of ±1600km.
Connections were made with the Beira and Cape Town
tides gauges
Generally, primary and secondary routes are made to
follow easy communication ways i.e. main roads in most
cases.
137
PRIMARY ROUTES




Primary circuits are first chosen and levelled and then
these are subdivided into secondary and tertiary routes.
They form the main framework of a country.
They run between fundamental benchmarks placed to the
junction of 3 or more lines of leveling.
Each primary circuit is allocated capital letter e.g. A.
138
SECONDARY ROUTES




Planned so that they are dependent on and are adjusted to
the primary framework.
The numbering of the secondary routes is based on the
primary circuits letters eg A9 i.e. the ninth route planned
in primary circuit A.
A secondary circuit formed by loops of secondary routes or
secondary & primary circuits is designated by 2 capital
letters e.g. AA, AB, AC i.e. all lines in the primary circuit
A.
Secondary routes are also terminated by fundamental
benchmarks.
139
TERTIARY ROUTES


Are dependent on and are adjusted to the primary and
secondary routes surrounding them.
The numbering of the routes is based in secondary circuit
number e.g. AF1
140
BENCH MARKS



1.
2.
3.

Bench mark is a permanent survey mark whose height is
known and from which levelling and heights are controlled.
Great care is needed in the siting of these marks on ground
that is thought to be stable and relatively free from vandalism.
There are 3 types:
Fundamental
Main
Intermediate
In each case a brass plug is set in concrete or rock.
141
BENCH MARKS



The quality and permanence of all levelling is heavily
dependent on the location and construction of the
bench marks used.
Wherever possible, marks should be sited where they
can reasonably be expected to be free from accidental
interference.
Because of the continuing problems of ground
movement, and the settling of structures, careful
consideration must be given to the location and type
of mark to be constructed, having regard to local soil
and rock characteristics, and the capital value and
permanence of the project involved.
142
FUNDAMENTAL BENCHMARKS








They are built on the terminal of primary and secondary leveling routes
at intervals of 65 to 95 km.
As the name implies, these benchmarks form the basic reference levels
of the network.
They are always built on live rock to minimize subsidence and on site
where they are most unlikely to be disturbed by future developments.
Live rock means that the rock appears to form part of the earth’s frame
and is not a detached boulder.
The BM consist of 3 hidden marks, the centre one being the
fundamental proper and designated CC.
The other 2 are AA and BB.
A public reference mark consisting of 18 inch concrete pillar and brass
plug is situated close to CC and AA, BB, CC are covered by cairns.
A fundamental benchmark is indicated from the number by an ‘F’ in the 143
middle e.g. 3F2, where 3 is the route and 2 would show that this is the
second fundamental benchmark in route 3.
144
MAIN BENCHMARKS



They consist of brass plug set on the top of a reinforced concrete
cylindrical pillar 30 cm diameter and 45cm above ground level.
These pillars will always be erected on live rock unless this is
unavoidable when they will be set in sound concrete foundation.
They are situated at 8km intervals and are numbered using the
convention <route no.> M < no. of mark> e.g. 3M46.
Intermediate Benchmarks



They are placed at 1.6 km intervals and they define the ends of
subsections of a leveling route.
The marks consist of brass plugs grouted in holes drilled in rock,
road bridge abutments or railway culverts.
When no stable foundation exists, they set in 30cm concrete cubes,
placed so that the top of the cube is flush with ground level.

The numbering convention is <route no.> /< no. of mark> e.g. 3/15.

A locality sketch should be drawn up of each benchmark as it is built.
145
INSTRUMENTATION






Levels
Tilting level with small circular bubble and main tabular
bubble e.g. Wild N3
Automatic Level
Which is roughly leveled by a circular bubble and residual
dislevelment is automatically corrected with a compensator e.g.
Zeiss N1002.
Staff/staves
Primary staves
The divisions are on invar material about 3m long either in cm
or ½ cm
2 scales are offset from one another to provide a check ad in the
field the staves are held vertical with steadying poles and
circular bubbles.
Secondary staves
Also 3m in length but with single scale
Divisions are in either cm or ½ cm.
146
INSTRUMENTATION (CONT…)
Parallel plate micrometer



This is attached to the level for primary and secondary
leveling.
The micrometer drum optically divide the intervals
between 2 successive graduations into 100 units thus for
staves with ½ cm divisions, a drum unit represent 0.05mm.
Tripods
They should be the rigid type.
147
FIELD PROCEDURES
ADJUSTMENTS, CHECK PRIOR TO TASK


-


Collimation – in the Two Peg Test, maximum error allowed is
1mm over 60m (i.e. 20 drum units).
Circular bubble – centre in one direction, rotate telescope
through 180o and check whether the bubble is still central.
If not, move the bubble half way back by turning the capstan screws
with a tommy bar
Tripod – there should be no play between the legs and head
Staves – check the staff bubbles with builder’s level placed
flat against the staff.
Test the staff’s frame for curvature.
The staff frame must be checked before commencing leveling
operations.
 This is done by stretching a thread over its length and the
curvature should not exceed 6mm.
 The staff should also be checked for the zero index error i.e. the
base not starting at 0.0.
 It can be checked by comparing its length with use of an invar
tape.


148
LEVELLING PROCEDURE
i.
ii.
iii.
iv.
v.
vi.
The tripod must be placed firmly midway between the forestaff and the back-staff using measuring cable.
One leg of the tripod is marked and placed facing the
marked staff whether in the forward or back position.
The telescope is pointed to the other station (or staff).
Start levelling with the circular bubble and then rotated
through 180o towards the marked staff.
The bubble should still be central
Keep the instrument under shade assuming the eye piece
is already focused, bring the staff into sharp focus
ensuring that there is no parallax.
149
THE SEQUENCE OF READING IS AS
FOLLOWS:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Read top and bottom stadia on the left hand scale (LHS) of
the marked staff.
Read the middle hair of the left hand scale.
Turn to the other staff and read the middle hair of the LHS.
Read top and bottom stadias
Read middle hair in the right hand scale (RHS).
Turn back to the marked staff and read the middle hair of
the RHS.
Subtract the bottom stadia reading from the top stadia
reading for each staff and find the difference between
forward and back shorts.
This must be within the tolerance allowed for the length of
the short.
Determine the offsets between RHS and LHS middle hair
readings for both staffs and ascertain that the 2 do not
differ by more than the specified maximum for the length of
short.
150
151
NO. OF OBSERVATIONS
Primary Leveling

8 observations taken per set up.

Secondary leveling
6 observations are taken per set-up to the bottom stadia,
center, center.

Tertiary leveling
4 observations are taken
152
PRECAUTIONS TO BE TAKEN DURING
PRECISE LEVELING



Errors that occur in precise leveling are either accidental or
cumulative.
Accidents are less important but cumulative errors must be
reduced to a minimum if accurate results are to be obtained.
The following precautions can be taken:
i. Instrument temperature – before starting a morning or
afternoon leveling, allow ten minutes for instrument to take up
the air temperature.
ii. Shading the level – the level must at all times be shaded from
the direct rays of the sun by means of an umbrella.
iii. Distances between staffs and level – the level must always be
set in the middle between 2 staves irrespective of the leveling 153
order. This helps in the elimination of collimation errors.





The lowest permissible centre hair reading – this should
not be lower than 0.5m (meant to remove effects of
refraction).
Tripod – the mark on the tripod must always point to the
marked staff. This eliminates any errors arising as a result
of the tripod legs tending to settle in one direction during
observing.
Checking circular bubble – The bubble must be
checked that it is in the centre of the etched circle before
taking any reading and immediately after.
Elimination of zero error – the commencement and
termination of a subscription of leveling must be with the
staff being held on the opening and closing BM. This means
the subsection must be divided into an even number of setups (any zero error is eliminated through this process).
Weather and atmospheric conditions – operations
cannot be carried out in strong windy and work must be
discontinued when observing conditions are not favourable.
Shimmer must not be there on staves.
154
TOLERANCES

1.
2.
3.
Each substation between intermediate BM is leveled in
both directions (direct and reverse). The limits for the
difference are:
Primary ⎷(D km), answer in mm
Secondary 7.5 ⎷D km, answer in mm.
Tertiary 15 ⎷D km, answer in mm.
155
CONNECTION OF BM’S TO TRIG STATIONS



1.
2.
To provide control for trig heights, leveling or trig heighting
connections are required to trig stations within a few kms of
the route and at intervals of ±13km.
The type of connection depends on the distance of the trig and
the topography.
There are 3 possibilities, which are:
You can perform spirit leveling. This is performed to the
same order as the parent route, leveling is done in both
directions.
Triangulation where 2 points are selected and leveled over
in the course of normal route leveling. A complete set of 2
arcs of horizontal angles and 2 sets of verticals are observed
at each of the 3 points. The base line distance say a-b is
measured twice in opposite directions.
156
EDM HEIGHTNING



Where 2 points (one or both) may be BMs are leveled.
EDM slope distance and vertical angles are observed from
pegs 1 or 2.
Vertical angles are also observed at the trigs back to the
pegs (1&2).
157
THE DETERMINATION OF MEAN SEA
LEVEL



The height of MSL is required as a basis for nation height
networks in order to obtain a measurement of the average
elevation of the sea, a continuous record of its varying levels is
required for a long period of time.
The period of observations should cover an integral no. of
lunations. And unless it extends to at least a year, the effects of
the solar components will not be truly represented.
The empowerment of a self registering tide gauging is essential
and its zero must be connected and periodically checked by
precise leveling to a permanent BM.
158
DEPARTURE OF MEAN SEA LEVEL FROM A
LEVEL SURFACE

1.
2.
3.
4.
MSL is only approximately an equi-potential surface for
the following reasons:
Sources of water (rain, rivers, melting ice bags), do not
coincide with areas of evaporation.
Varying water density owing to varying temperature and
salinity.
Overlying atmosphere with varying pressure
Wind applying horizontal force because of these
inequalities, ocean correction arrive seeking to restore
equilibrium but with a time lag.
159
TRIG HEIGHTING

It is a technique of finding height difference between
stations by just observing vertical angles and distances.

Distances can be measured by use of an EDM.

Distance from coordinates is termed spheroidal distance.

Trig heighting is however less accurate as compared to
spirit leveling but the accuracy achieved enables controls to
be used for topographical mapping.
160
METHODS OF TRIG HEIGHTING
There are 2 methods:

-

Direct method – here the difference of elevation is
computed from the vertical angles measured at one of the
stations only.
Knowledge of the refraction coefficient (k) is required so as
to cater for the effects of curvature and refraction.
Reciprocal method – this method can be split into 2 i.e.
reciprocal trig leveling or simultaneous reciprocal trig
leveling.
161
RECIPROCAL METHODS





Reciprocal trig leveling
Vertical angles are observed from each station to the other
but not at the same time.
The angle of refraction is taken as being equal for both
stations and is eliminated in the reduction.
Simultaneous reciprocal trig leveling
Observations are taken at each end of the line at the same
time so as to eliminate effects of curvature and refraction.
In reciprocal observations, vertical angles have to be
corrected for the effects of the variation of the target and
the instrument height (eye object correction).
Additional measurements
On top of the vertical angles and slope angle and distance,
respective height of instrument also needs to be measured.
162
BASIC CONCEPT
163
BASIC CONCEPT





Consider 2 points A & B whose difference in height is to be established
using trig leveling method:
It can be seen that ∆h = S Sin ∝
∆h = s Cos z
∆h is also calculated using D tan ∝
∆h = D tan ∝
∆h = D cot z
The difference in height or elevation
∆H between points A and B is given by ∆H = hi + ∆h - ht
For reduced levels (RL)
RL = RLA + ∆HAB
= Rla + hi + ∆h - hT
where:
hi –height of instrument
ht - height of target
∆h - vertical distance
∆H – height difference
164
NB. If the distance between the 2 points A and B is very
large, the effect of curvature and refraction must be
considered hence the formula will become:
RLB = RLA + hi + ∆h – hi + (c-r)
(c-r) = s2 (1-K) /2R
Where K is the coefficient of refraction
R – radius of the earth
s – horizontal distance
165
CONVERSION FROM SPHEROIDAL TO
HORIZONTAL
s = So (1 + hm/R)
where
So – spheroidal distance
s = horizontal distance
hm – mean height
∆H = hi + ∆h – ht +(c-r) where ∆h = D tan ∝
∆H = hi + So (1+ hm/R)tan∝ + s2(1-K)/2R - ht
166
EXAMPLE – THE FOLLOWING ZENITH ANGLES WERE OBSERVED
FROM POINT P TO 3 TRIGS 65/T, 30/T AND 51/T FROM THE DATA
GIVEN BELOW, DETERMINE THE REDUCED LEVEL OF P.
To
CL
CR
Signal
Spheroidal
RL
65/T
88 46 34(TOV)
271 13 58
1.609
4185.38
1568.541
30/T
90 00 10(TOV)
269 59 34
1.650
5275.61
1479.56
TOP – top of pillar
TOV – top of vein
Mean height of area = 1500m
hi = 1.5m
R = 6370km
K = 0.12
51/T
90 00 42(TOP)
270 00 11
4350.92
1480.99(TOP)
167
ANSWER
168
DATUMS







Geodetic datum- there are many different ellipsoids on which
positions may be expressed.
The size, shape and positioning of the ellipsoidal reference
system with respect to the area of interest is largely arbitrary
and determined in different ways around the globe.
There are various parameters which define a geodetic datum
namely (refer to research/ assignment).
Geodetic datums define the size, shape of the earth, origin and
orientation of the coordinate system.
Modern geodetic datums range from flat earth models used for
plane surveying to complex models used for international
applications which completely describe the size, shape,
orientation, gravity field and angular velocity of earth.
While cartography, surveying, navigation and astronomy all
make use of geodetic datums, the Science of geodesy is the
central discipline for the topic.
Referencing geodetic coordinates to the wrong datum can
result in positional errors of several hundreds of meters.
169



Different nations and agencies use different datums as a
basis for coordinate systems hence the diversity of datums
required.
Careful datum selection and careful conversion between
coordinates on different datums.
Some of the examples include;
1.
2.
3.
Cape datum
WGS84
Clarke 1880 modified

Southern Africa particularly Zim, utilizes Clarke 1880
modified
South Africa is in the process of adopting WGS84.

The science of geodesy is the central discipline of the topic.

170
COMPUTATIONAL SURFACES


Include the earth’s surface itself, ellipsoidal surface and
geoidal surface.
Topo surface is very irregular so unsuitable for any
meaningful computations

Geoidal surface is only suitable for height determination.

Ellipsoidal surface is accurate for planimetric fixing.
171
HEIGHT SYSTEMS


Height of a point is its geometrical distance above a datum
(Geoid/ellipsoid), and it is measured along the gravity vector
or along an ellipsoidal normal passing through the point in
question.
There are 3 main height systems namely:
1.
Orthometric
2.
3.



Dynamic
Normal heights
A problem common to all the three height systems in some
countries such as Zimbabwe, is the lack of surface gravity
data.
The problem was by-passed by using normal gravity ⋎ instead
of g.
Height differences based on normal gravity do not differ
significantly from proper values provided only adjacent BM’s
are considered.
172
HEIGHTS
 ORTHOMETRIC
The distance between the geoid and a point on the Earth’s surface
measured along the plumb line.
 GEOIDAL UNDULATION
The distance along a perpendicular from the ellipsoid of reference
to the geoid
 ELLIPSOID
The distance along a perpendicular from the ellipsoid to a point
on the Earth’s surface.
 DYNAMIC
The distance between the geoid and a point on Earth’s surface
measured along the plumb line at a latitude of 45 degrees
173
HEIGHTS BASED ON GEOPOTENTIAL
NUMBER (C)

Normal Height



 = Average normal gravity along plumb line
Dynamic Height

45 = Normal gravity at 45° latitude
Orthometric Height

H* = C / 
H
dyn
= C / 45
H = C / gm
g = Average gravity along the curved plumb line
Geopotential number is the number of the level surface on which it
lies, where the number is numerically equal to the amount of work
required to raise unit mass to it from mean sea level.
174
ORTHOMETRIC HEIGHTS




This refers to the linear distance measured along the
gravity vector from the point to the equi-potential surface
used as a reference datum (Geoid).
An equi-potential surface in this case is one on which all
the points have the same gravitational potential,
Since gravity varies with latitude.
Orthometric height is the actual height of a point above the
geoid measured in metres or other linear units.
175
ORTHOMETRIC HEIGHTS (CONT…)
Metrication is done by use of mini-value of gravity along
the vector, mini-gravity is computed from the surface
gravity of the point to be heighted in a process called
downward continuation of point gravity data to the datum
*NB: Orthometric height on same equi-potential surface can
only be the same if both measured height and mini-gravity
are the same.
 One problem in measuring orthometric heights is the
problem in downward process to the geoid.
 A line of constant orthometric elevation is supposed to be
parallel to the MSL surface.
 However a problem arises because above the geoid level
surfaces are generally not parallel to the geoid since the
vertical or plumbline is curved
 This results in error with levelling circuits failing to close
by significant amounts.

176
ORTHOMETRIC HEIGHTS (CONT…)

Ѳ is inclination between ab and AB
177
ORTHOMETRIC HEIGHTS (CONT…)





𝜹O = 𝜹M + Ѳl
OR
Where 𝜹O – is orthometric height difference
Ѳl = orthometric correction to be applied
𝜹M is the measured height difference. Using the approximation g = ⋎h,
where
⋎h = 978 (1 + 0.0053 Sin2ф – 2h/R) gals
With ф = lattitude, R = mean earth radius, we get
dg/dф = 978 (0.00530sin 2ф) gals per radian
= 978 (0.00530sin 2ф) ms-2 rad-1
hm – mean elevation of 2 points above MSL




Substituting, we finally get
𝜹O =𝜹M – hm (𝜹ф)’’ (0.0053 sin2фm) sin 1’’
This is what may be called “spheroidal” orthometric height since actual178
gravity has been replaced by normal gravity.
It is the system that has been used in Zimbabwe
TRUE ORTHOMETRIC HEIGHTS



Where C = geopotential number of P at h = Aa.
It is practically impossible to determine gm because g along
the plumbline is not known since the density distribution
within the earth is not known.
Numerous approaches to approximate gm each lead to a
specific kind of orthometric height attributed to its
proponent e.g Helmert.
179
Leveled Height vs. Orthometric Height
 h = local leveled differences
H = relative orthometric heights
B
A
HA
Topography
 hAB =  hBC
C
HAC  hAB + hBC
HC
Observed difference in orthometric height,
H, depends on the leveling route.
DYNAMIC HEIGHTS

Dynamic heights are measured by the work required to
raise a unit mass from MSL to the point.
The geopotential number of any point is the number of the
level surface on which it lies, where the number is
numerically equal to the amount of work required to raise
unit mass to it from mean sea level.

Geopotential no. c = ∫gdh




Dynamic height = HD = C/gr
where gr = reference gravity
g is the variable acceleration due to gravity. The system
ensures that points on the same level surface have
identical heights known as geopotential numbers, whose
units are called geopotential units (GPU).
1 GPU = 1 kgal metre = 1 00 000 cm2 s-2 =10 m2s-2
181
DYNAMIC HEIGHTS


Since gravity varies with latitude, a line of constant
dynamic elevation is parallel to mean sea level only when it
lies along a parallel of latitude.
This type of height system has its main application in
hydrological projects as it guarantees water flows from a
higher to lower heights.
182
RELATIONSHIP BETWEEN ORTHOMETRIC
AND DYNAMIC HEIGHTS




The difference between orthometric and dynamic heights
and the necessity for the orthometric correction may be
understood from the following consideration:
A and B are 2 points at sea level and C and D are 2 other
points which are vertically above A and B and lie on a
dynamic level or equi-potential surface DC
Owing to the variation of gravity with latitude and height
of station, the 2 equi-potential surfaces DC and AB tend to
converge towards the pole and to diverge from one another
towards the equator, hence the height BC is less than
height AD but dynamically height D = height C.
Although the dynamic heights are equivalent, the
orthometric elevations are not equal.
183
184
ELLIPSOIDAL, ORTHOMETRIC AND
DYNAMIC HEIGHTS




h = N + H cos §
However if § is small, it can be ignored, hence h = N+H
h – ellipsoidal height
H – orthometric height
N – Geoid-ellipsoidal separation (geoidal height).
It has to be noted that if the orthometric height at a point
is known, its equivalent dynamic height can be derived
from the expression
h – is Dynamic height
185
QUESTION

Calculate the dynamic height at a point on the
equator whose orthometric height is 1010m, also
calculate the geoidal height.
Answer:
 Dynamic height = 1007.33m
 Geoidal height (N) = 2.67

186
NORMAL HEIGHTS




Reference is made to a determinable reference surface
called Quasi-Geoid.
Normal heights are geopotential numbers which have been
scaled using mini-gravity of the ellipsoidal normal.
Normal heights are easy to transform to height above the
ellipsoid since geometrical distance above the ellipsoid are
easy to establish.
The normal correction can be completed without making
assumptions on the mass distribution inside the earth.
187
NORMAL HEIGHTS (CONT…)



Molodensky suggested normal heights HN , defined as
HN = C/⋎m .
where ⋎m is the normal equivalent of gm i.e. the mean normal
gravity along the plumbline of P.
While gm is computed from downward continuation of normal
gravity ⋎o on the geocentric reference ellipsoid, using accurate
value for the vertical gradient.
Normal heights refer to a determinable reference surface called
the quasi-geoid.
188
QUESTIONS????
SURVEYING III
COORDINATE SYSTEMS
&
PROJECTIONS
Juliana Useya
Department of Geoinformatics and Surveying
DEFINITION


1.
2.
A coordinate system is a unique mathematical system of
describing the position of a feature on the earth's surface.
A coordinate system helps in answering the questions:
What is where?
Where is what?
191
REQUIREMENTS OF A COORDINATE SYSTEM



A coordinate must have:
Orientation
Reference axis
I.
II.



Primary axis (which is the axis to which the directions are
referenced from)
Secondary axis (which is an axis perpendicular to reference
axis passing through the origin)
Origin at the intersection of the two axes
Scale
Units
192
TYPES OF COORDINATE SYSTEMS
1. Global/Geographical coordinate system
 Is also referred to as Spherical coordinate system
 This system uses latitude (ф) and longitude (λ) to describe
the position of features.
Greenwich meridian (0⁰λ)
Equator (0⁰ latitude)
Notation: P (фp ; λp)
Units are either degrees or
radians
193
PROBLEMS


The earth is not a perfect sphere.
Mathematical computations are complex (spherical
trigonometry).
SOLUTION:

Project the curved surface of earth onto a plane using MAP
PROJECTIONS.
194
2. PLANE COORDINATE SYSTEM:
THE POSITION OF A FEATURE IS DESCRIBED BY EITHER :
The distance from the 2 axes
The system is referred to as the
GRID/RECTANGULAR/
CARTESIAN COORDINATE
SYSTEM
Angle and distance from the origin
to the point.
The angle is measured from the
primary axis to the line joining
the origin and the point.
The system is known as POLAR
COORDINATE SYSTEM.
195
POLAR COORDINATE SYSTEM


Is difficult to use for simple distance calculations
But it is useful in some survey methodologies
tacheometry surveys.
e.g
196
GRID COORDINATE SYSTEM

Is mostly used because the calculations are much simpler.
2.
In Zimbabwe, we use two different Grid coordinate
systems:
GAUSS COORDINATE SYSTEM
UNIVERSAL TRANSVERSE MERCATOR (UTM)

Both are conformal projections, they preserve shape.

1.
197
THE GAUSS COORDINATE SYSTEM




The origin is at the intersection of the equator and central meridian.
The system is a 2⁰ belt system (each coordinate system covers a belt
which is 2⁰ wide in longitude).
The origin is TRUE (0;0).
Central meridians for Zimbabwe are 25⁰E, 27 ⁰ E, 29 ⁰ E, 31 ⁰ E & 33
⁰ E longitude.
198
COORDINATE SYSTEM USED FOR
CADASTRAL SURVEYING IN ZIMBABWE



The primary reference axis is TRUE SOUTH i.e. 0⁰
direction is true south, hence North is 180⁰.
Y coordinates increase +ve WEST
X coordinate increase +ve SOUTH
199
UNIVERSAL TRANSVERSE MERCATOR
(UTM)




The earth is divided into 60 zones, each zone being 6⁰ wide.
The origin is false (to avoid negative coordinate values of any
part of the system).
The northern hemisphere, the origin is assigned coordinates
(500 000;0).
The southern hemisphere, the origin is assigned coordinates
(500 000;10 000 000)
200
CENTRAL MERIDIAN FOR ZIMBABWE

Are 27⁰E and 33⁰E longitudes
201







Only 2 projections are used in Zimbabwe
which are UTM and GAUSS.
The following notes compare the 2 projections
The UTM and GAUSS are both transverse mercator which has
the property of conformality (preserve shape).
The transverse mercator uses a projection cylinder whose axis
is imagined to be parallel to the earth equatorial plane and
perpendicular to its axis at spin.
The UTM belt is 6⁰ wide, 3⁰ either side of the central meridian
which in the case of Zimbabwe coincides with 27⁰ and 33⁰ East.
The Gauss belt is 2⁰ wide with central meridian 25⁰, 27⁰, 29⁰,
31⁰, 33⁰.
The scale for Gauss is unit (1) on central meridian and 0.9996
for UTM.
UTM is used for mapping and Gauss for cadastral beacon
coordinates
202
QUESTIONS????
SURVEYING III
COORDINATE
TRANSFORMATIONS
Juliana Useya
Department of Geoinformatics and Surveying
INTRODUCTION





Transformation of coordinates refers to the conversion from one
coordinate system to another e.g. transforming coordinates from
a local system to a national system e.g. WGS84 to Clarke 1880
modified.
To perform any transformation there is need to establish three
basic parameters:
Rotation (Swing) – about the three axes Ѳx,Ѳy and Ѳz in order to
render the axes of the systems involved parallel.
Translation – of the origin ) which would involve shifts in X, Y
and Z i.e. ∆X, ∆Y and ∆Z.
Scale factor (S) – to equalize the scales of the different coordinate
systems.
205
INTRODUCTION (CONT…)
At least two points should have coordinates on
both systems.
 Swing (Ѳ) = old direction (local system) – new
direction (national system)
 Scale factor (s) = distance on new system/distance
on old system


206
EXAMPLE

Given the following information determine the
coordinates of C and D on the new system.
Old
New
Point
Y
X
Y
X
A
-40076.90
+26 992.30
+376.96
+355.13
B
-39588.91
+27 409.79
+540.56
+486.21
C
-39832.02
+27 389.59
D
-39699.43
+27 437.12
207
SOLUTION

From Join calculation
Old system
New system
Direction AB
49° 27’ 07’’
51° 17’ 51’’
Distance
642.208798
209.635222
Swing (Ѳ)
-1° 50’ 44’’
Scale factor (s)
0.325
208
Name
Old
Y
X
A
-40076.90
+26992.30
∆old
+487.99
B
Station
New
Y
X
A
+376.96
+355.13
+417.49
∆new
+163.60
+131.08
-39588.91
+27409.79
B
+540.56
+486.21
∆old
-243.11
-20.2
∆new
-79.529
-4.036
C
-39832.02
+27389.59
C
+461.03
+482.187
∆old
+132.59
47.53
∆new
+113.758
+14.113
D
-39699.43
27437.12
D
+504.79
+496.29
∆old
-377.47
-444.82
∆new
-127.829
-141.58
A
-40076.90
+26992.30
A
376.96
355.13
0.00
0.00
Check (∑∆) 0.00
0.00
209
TYPES OF TRANSFORMATIONS
Stretch
 Shear
 Similarity
 Affine

210
QUESTIONS????
END OF MODULE
BEST WISHES FOR THE EXAM
THANK YOU!!!!
213
214
215