Horizontal control surveys

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

Error of closure: compass and transit rule

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.
Law of sines inside oblique triangle is used.
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.)

In geodetic surveying, the task is very similar. We want to fit locations of control points in such manner that the movements (either on angles or distances) are minimal.

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.

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