A typical construction project is developed and planned in the context of
coordinates. It is usual and convenient that such coordinates are derived from
a network of control points. These points are established by a
control survey and are up to tens of kilometres apart.
Fixed control points are usually built as control stations
(monuments) by government agencies. On the picture below is depicted
primary triangulation network in the Netherlands made up of 77 points (mostly
church towers). Considering how these reference points are being found,
it so no surprise that control stations are usually established on high
locations (contrary to benchmark points).
The old primary triangulation network in the
Netherlands made up of 77 points (mostly church towers). The extension and
re-measurement of the network is nowadays done through satellite
measurements
Horizontal control network may be established by one of the following methods:
traversing,
triangulation,
trilateration,
GPS,
photogrammetry.
Traversing and triangulation (measuring angles of triangles) are further
discussed in this chapter. These are traditional methods to find coordinates of the stations and are time
consuming.
Total stations (electronic equipment) can be essentially helpful for
trilateration: time and personal expenses can be reduced by using this method.
Trilateration stands on the system of triangles, where
only lengths are measured. The other method, further reducing time expenses, is
GPS. Costs of GPS equipment are high. But affordability will
likely change in the future, and GPS might become more efficient than control
monuments.
Traverses
A traverse survey measures distances and angles (courses) between the
points.
These points can serve as control stations. Other, less precise
measurements, can be taken from control stations then. Traversing is used usually
for smaller areas or on areas with many obstacles. The method is
appropriate for land and property surveys as well.
Types of traverses
Traverses are classified as open and
closed. By closing the loop, we have the ability to check result
against errors and blunders. Unclosed traverse can not be
checked, are not recommended and need to be taken two times to avoid
blunders.
What is being assumed or taken from other survey is depicted in red color in diagrams below.
Types of traverse: open and closed traverse
For long traverses, checks on the measured horizontal angles can be
obtained by making periodic astronomical azimuth observations.
Check of traverse against astronomical azimuth observation
Control traverse station
Control stations, which are used to fix control points, must mark
the point position clearly.
Can be made from wooden stakes with a nail
or concrete monuments nearly aligned with ground.
Station's witness points
The station has its accessories called witnessing points. These are
reference points easily visible around station. They are most useful when
conditions in the area are deteriorated by snow, floods, leaves, soil, ...
Either they can be built or current monuments can be used (e.g. old tree). Approximate locations of 3
to 4 witnesses from witnessed control point are recorded into protocol.
Measurements
Loop traverses should be measured counterclockwise.
Because it is most convenient to read interior angles
from left to right. It is a fast method and blunders will be avoided
likely.
Data reduction (for traverses)
The process of converting all the distances and angles into
coordinates is called data reduction.
Error of closure
It is not expected that data taken from a survey will close exactly.
If the error on closure is acceptably small, then data are improved by
corrections calculated from error.
The method is explained in text further.
If the error on closure exceeds acceptable limit, survey has to be repeated.
Horizontal control survey: error of closure
Horizontal control survey: compass and transit rule to recover from an error of closure
When a traverse is closed to the BM2, an error of closure is observed during the process of data reduction. Then two basic rules are being used to distribute the error: 1—compass rule: error is distributed to the courses based on their length. Since course BM1-A is longer, most of the correction belong to that course 2—transit rule: error is distributed to the courses based on latitude and longitude components of each course; the course BM1-A runs from West to East, thus contains only longitude component and vice versa; note that the courses do not change direction: transit is an equivalent to theodolite; assuming that accuracy of angles is higher than accuracy of distances, then we prefer not to change the directions
Traverse closure computation
Computation will be demonstrated on the example below. There are several steps to conduct:
Fix angles measured for corrections.
No measurement
is perfect, and after closing the path, it will be found that sum of all interior angles
has some error from expected sum.
Evaluate azimuths of courses.
From field work, we have relative
angles, which are not related to widely used coordination system.
Evaluate latitudes and departures of courses.
This is not difficult if we have azimuths and lengths of courses.
Fix latitudes and departures of courses for
corrections.
Measurement of distances is imperfect as well. We have to
find the errors and divide them back as corrections.
Then final transformation into global coordination system can be made.
That is the most easy step, coordinates will be moved according to location of known point (not in our example).
Example for traverse closure computation
Sum of interior angles
The sum of interior angles in any loop traverse is equal to (n - 2) × 180°, n is count of sides:
Sum of interior angles in polygon
We find out sum of unadjusted angles, error and then error will be applied back as a correction.
Station
Angle taken
Angle adjusted
A
64°53'30"
64°53'00"
B
206°35'15"
206°34'45"
C
64°21'15"
64°20'45"
D
107°33'45"
107°33'15"
E
96°38'45"
96°38'15"
Sum
537°180'150"
537°178'120"
540°02'30"
540°00'00"
Error
02'30"
Per angle
00'30"
Example for traverse closure computation
Compute azimuths (for traverses)
We want to compute coordinates of all stations, and we are being used to
work with rectangular coordinates (or latitude/longitude to be more exact).
The
task will be first to establish semi-polar coordinates.
Because polar coordinates
follow a nature of data taken from field work.
In the next step, semi-rectangular coordinates
will be derived.
Azimuth is our gate to use polar system.
Course
Azimuth
Length (L)
Δx
Δy
DE
98°17'45"
971.26
961.10
-140.14
EA
14°56'00"
783.32
201.86
756.86
AB
259°49'00"
690.88
-680.00
-122.15
BC
286°23'45"
616.05
-591.00
173.89
CD
170°44'30"
677.97
109.08
-669.14
DE
98°17'45"
∑ =
3739.48
1.04
-0.66
Example for traverse closure computation
Compute rectangular coordinates (for traverses)
In the table above there are errors at distances found.
These will be applied back as corrections.
If path from start point D is followed according to the measured
values, the endpoint D will not be reached. Position 1.04 m right and -0.66 m
down from D will be the final point. Because closing point is shifted to the
right, if the endpoint of each course is moved a little to the left then, error
is being cancelled. Two basic methods to divide error into particular courses
can be used. These methods are called the compass rule
and the transit rule.
The compass rule
This method computes and divides error for each course
according to its length.
For example in case of
DE course, we apply 26.9 % (i.e. 971.26 / 3739.48) of errors Δx, Δy.
Course
C.
Δx
Δy
corr_{x}
corr_{y}
Δx_{final}
Δy_{final}
DE
961.10
-140.14
-0.27
0.17
960.83
-139.97
EA
201.86
756.86
-0.22
0.14
201.64
757.00
AB
-680.00
-122.15
-0.19
0.12
-680.19
-122.02
BC
-591.00
173.89
-0.17
0.11
-591.17
174.00
CD
109.08
-669.14
-0.19
0.12
108.89
-669.02
∑ (check)
-1.04
0.66
0.00
-0.01
The compass rule is appropriate for surveys, where angles were taken with
lower precision than distances.
The transit rule
First only latitudes are used to fix the errors within latitudes. Then only departures are used to fix the errors within departures. For each
course its latitude is taken. It is compared to length of all latitudes. Proportional
correction is taken (from error found on latitudes) and applied. Analogically for departures
as well.
In example error on latitudes was found to be 1.04 m, therefore -1.04
m has to be divided back as a correction. Lengths of all latitudes
were measured as 2543.04 m. Course DE has latitude of 961.10 m. We will apply
37.7 % (i.e. 961.10 / 2543.04) of error ∑Δx to latitude of course DE.
Course
C.
Δx
Δy
corr_{x}
corr_{y}
Δx_{final}
Δy_{final}
DE
961.10
-140.14
-0.39
0.05
960.71
-140.09
EA
201.86
756.86
-0.08
0.27
201.78
757.13
AB
-680.00
-122.15
-0.28
0.04
-680.28
-122.11
BC
-591.00
173.89
-0.24
0.06
-591.24
173.95
CD
109.08
-669.14
-0.04
0.24
109.04
-668.90
∑
2543.04
1862.18
-1.03
0.66
0.01
-0.02
The transit rule is appropriate for surveys, where angles were taken with
greater precision than distances.
Triangulation
Triangulation is a method based on measuring angles
using theodolite on a network of triangles.
To convert angles into coordinates, some
base distance have to be taken too. To have a control for
closure, check base distance is measured as well. More
distances can be taken to strengthen accuracy of network. The distance of
control points measured depends on purpose and can vary up to tens of
kilometres. According to demanded accuracy, instrument with reading of 0.1"
might be requested. The result is a network of located control
points.
Data reduction for triangulation (simple method)
Data reduction can be conducted in many ways. Most
advanced approaches employ the least-square method.
This mathematic method follows an idea that
necessary corrections of data from the field should be done as small as possible.
Then best accuracy, which complies with all boundary conditions, will be
provided. To use this method, advanced mathematical background to understand
and find all geometry conditions is needed and is further discussed
here on the end of the chapter.
The simplified method of data reduction will be demonstrated in several steps on simple example.
Firstly, corrections of interior angles within triangles are made.
That
means sum of interior angles is corrected to be of 180 °.
Computation start from known distance at the base and lengths of all courses are evaluated.
Finally, all distances are rescaled to
fit into bases.
Finally, comparison of
distance at check base is being made, and all lengths are rescaled then. If the
computed check base distance is 1000.20 m while measured check base is 1000.00
m, then all distances are rescaled to the average 1000.10 m check base.
Triangulation data reduction example
100ft Ramsden Chain used in his survey (around 1784) to measure distances
In the beginnings, triangulation employed chains to measure both the base and
the check base. These lines were usually short and the triangulation relied on very
accurate measurements of angles. Nowadays EDM is used to measure distances of
up to tens of kilometers.
Trilateration
is a simplified method of triangulation. Time consumption of the whole process can be reduced:
In some cases (e.g. using EDMI on long distances)
taking only distances between control points might provide
sufficient or even better accuracy.
All the
general rules for triangulation can be applied to trilateration. Task can be
transformed to triangulation. Because it is possible to evaluate interior
angles from sides of triangles, which are known.
Lengths measured with EDMI need to be fixed for slope and earth curvature.
The least-square method for data reduction
The data reduction explained above is feasible only in cases of simple
surveys.
In general, geometry for triangulation and trilateration is usually overdetermined,
and conventional methods fail to find a proper solution.
Such
complication of data reduction is an advantage since overdetermination
tells us that the network is strong; thus we can dig more accurate
results.
(above)—The determined networks of triangles: all the lengths/angles can be found straight-forward using a basic trigonometry math.
(below)—The overdetermined network of triangles: the network is stronger than the above network. That is a good point indeed, but traditional methods fail to find the best solution from the survey's data even in the case of the network as simple as depicted. Because the points belongs not to a single location; it depends where the computation starts.
Proposed system of triangles
The least-square method
Let us consider an experiment where a cube from concrete is exposed to the
compression test in order to find out the strength of the material. For such
purpose, not one, but ten specimens are prepared and tested.
While the size of each specimen can be considered the same, the observed
force can vary from 62 MN to 75 MN for a given size.
Compression test on a concrete specimen
If we draw observed force for a variety of specimens, we might get a chart resembling these below:
(above)—Data fitting along the curve.
(below)—The best fitting curve for data can be found by
least-square method. That means that the sum of
squares of distances from curve has to be minimal. (The vertical
offset approach is considerably simpler for practical usage and usually
provides an approximation of requested quality.)
That employs quite a lot of mathematics and geometry
knowledge. Today, numeric approach for least square method is feasible to solve
such tasks (software is available).
Demonstration of a computer program which implements the least-square method to find out locations of stations from field data.
1—Let us demonstrate on a case of trilateration with stations A to E.
In blue color are measured values. Blue line is considered to be fixed. This
is an ideal case with no errors involved. In a form there is a list of 5 points
A-E entered with their rough coordinates, fixed points are in red. Below list of points are
data taken from the field.
2—Let us simulate a systematic error. In the picture, error is simulated
only on side CD and is exaggerated. In a real survey, the imperfection
will be spread thoroughly over all measurements.
3—The impact of the error is depicted. Solving such a task will bring
us into troubles. One point appears to belong into two locations, depending on
where the calculations starts. We may decide to put point C into the middle of
illusive locations, but we have then to enlarge BC from 1414.2 to 1444.8. This is far
from the optimal solution. We call the algorithm to adjust data from the survey.
4—The adjusted coordinates have been entered by algorithm into the upper part of the
window. The picture is redrawn. The algorithm has computed changes to measured
values. These are small and are spread thoroughly over all measurements. This is
the optimal solution.