QUESTION: DESCRIBE THE SOURCES OF ERROR IN LEVELING

            DESCRIBE THE SOURCES OF ERROR IN LEVELING

            Many sources of error exist in levelling and the most commonly met in practice are discussed. Firstly, one of the sources of error is errors in the equipment which is collimation error. This can be a serious source of error in levelling if the sight lengths from one instrument position are not equal, since the collimation error proportional to the difference in sight length. The line of collimation should be parallel to the line of sights. 

Hence, in all types of levelling, sights should kept equal, particularly back sights and fore sights. Before using any level it is advisable to carry out a two-peg to ensure that the collimation error is as small as possible. Other than that, compensator not working. The function of compensator is to deviate the horizontal ray of light at the optical center of the object lens through the center of the cross hairs. This ensure that line of sight viewed through the telescope is horizontal.  

If the reading changes to a different position each time the footscrew is moved or thr instrument tapped, the compensator is not working properly and the instrument should be returned to the manufacturer for repair. Parallax  also one of error in the equipment. Parallax must be eliminated before any readings are taken. Parallax is occur when the image of the distance point or object and focal plane are not fall exactly in the plane of the diaphragm. 

To eliminate parallax, the eyepiece is first adjusted until the cross hairs appear in sharp focus. Then, defects on the staff  which is the incorrect graduation staff cause the zero error. This does not effect height differences if the same staff is used for all the levelling but introduces errors if to staves used for the same series of levels. When using a multisection staff, it is important to unsure that it is properly extended by examining the graduations on either side of each joint. The stability of tripods should also be checked before any fieldwork commences .

 

                     Secondly, field errors also source of error. The example of field errors is staff not vertical, failure to hold the staff vertical will result in incorrect readings. The staff is held vertical with the aid of a circular bubble. At frequent intervals the circular bubble should checked against plumb line and adjusted if necessary. Another example of field errors is unstable ground. When the instrument is set up on soft ground and bituminous surfaces on hot days, an effect often overlooked is that the tripod legs may sink into the ground or rise slightly while readings are being taken.This alters the height collimation and therefore advisable to choose firm ground on which to set up the level. 

After that, handling the instrument and tripod as well as vertical displacement, the HPC may be altered for any set-up if the tripod is held or leant against. When levelling, avoid contact with the tripod and only use the level by light contact through the fingertips. Then, instrument not level is also the field errors. For automatic levels this source of error is unusual but, for tilting level in which the tilting screw has to be adjusted for each reading, this is common mistake. The best solution is to ensure the main bubble is centralised before and after reading.

 

                 Thirdly, source of error is the effects of curvature and refraction on levelling. The effect of atmospheric on the line of sight is to bend it towards the Earth’s surface causing staff readings to be too low. This is variable effect depending on atmospheric condition but for ordinary work refraction is assumed to have value 1/7 that of curvature bit is of opposite sign. The combined and refraction correction is c + r = 0.0673 D². If longer sight lengths must be used, it is worth remembering that the effects of curvature and refraction will cancel if the sight length are equal. But, curvature and refraction cannot always be ignored when calculating heights using theodolite methods.

 

                  Lastly, source of error is reading and booking error and also weather conditions. Source of reading error is the sighting the staff over too long a distance, when it becomes impossible to take accurate readings. It is , therefore, recommended that sighting distances should be limited to 50m but, where absolutely unavoidable, this may be increased to maximum of 100m. For weather conditions, when it windy will cause the level to vibrate and give rise to difficulties in holding the staff steady. In hot weather, the effect of refraction are serious and produce a shimmering effect near ground level. The reading cannot be read accurately.

QUESTION : EXPLAIN BASIC RULES IN PRACTICE WHEN CONDUCTING A LEVELING

1.     EXPLAIN BASIC RULES IN PRACTICE WHEN CONDUCTING A LEVELING

    

                Levelling is the process of measuring the difference in elevation between two or more points. In engineering surveying, levelling has many application and is used at all stages in construction projects from the initial site survey through the final setting out. In practice, it is possible to measure heights to better than a few millimeters when levelling  this precision  is more than adequate for height measurement on the majority of civil engineering project.

The basic rules in practice when conducting a levelling fieldwork should be adhered to if many of the sources of error are to be avoided. Levelling should always start and finish  at points of known reduced level so that misclosures can be detected. When only one bench mark is available, levelling lines must be run in loops starting and finishing at the bench mark. 

Where possible, all sights length should below 50m. The staff must be held vertically by suitable use of a circular bubble or by rocking the staff and notong the minimum reading. Backsight and fortsight length should be equal for each instrument position. For engineering application, many intermediate sight readings may be taken from each set- up. Under this circumstances it is important that the level has no more than a small collimation error. 

Reading should book immediately after they are observed and important readings, particularly at change points, should be checked. The rise and fall method of reduction should used when heighting reference or change  points and the HPR method ( height of collimation) should be used for contouring , sectioning and setting out applications.

Lesson Note On Solving plane triangles

Solving plane trianglesEdit

Standard notation for a triangle

A general form triangle has six main characteristics (see picture): three linear (side lengths a, b, c) and three angular (α, β, γ). The classical plane trigonometry problem is to specify three of the six characteristics and determine the other three. A triangle can be uniquely determined in this sense when given any of the following:

• Three sides (SSS)
• Two sides and the included angle (SAS)
• Two sides and an angle not included between them (SSA), if the side length adjacent to the angle is shorter than the other side length.
• A side and the two angles adjacent to it (ASA)
• A side, the angle opposite to it and an angle adjacent to it (AAS).
• Three angles (AAA) on the sphere (but not in the plane).

For all cases in the plane, at least one of the side lengths must be specified.

 If only the angles are given, the side lengths cannot be determined, because any similar triangle is a solution

Trigonomic relations

The standard method of solving the problem is to use fundamental relations.

Law of cosines

${\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha }$

${\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta }$

${\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma }$

Law of sines

${\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}}$

Sum of angles

${\displaystyle \alpha +\beta +\gamma =180^{\circ }}$

Law of tangents

${\displaystyle {\frac {a-b}{a+b}}={\frac {\tan[{\frac {1}{2}}(\alpha -\beta )]}{\tan[{\frac {1}{2}}(\alpha +\beta )]}}.}$

There are other (sometimes practically useful) universal relations:

 the law of cotangents and Mollweide's formula.

NotesEdit

1. To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the cosine value unambiguously determines its angle. On the other hand, if the angle is small (or close to 180°), then it is more robust numerically to determine it from its sine than its cosine because the arc-cosine function has a divergent derivative at 1 (or −1).
2. We assume that the relative position of specified characteristics is known. If not, the mirror reflection of the triangle will also be a solution. For example, three side lengths uniquely define either a triangle or its reflection.

Three sides given (SSS)Edit

Three sides given

Let three side lengths a, b, c be specified. To find the angles α, β, the law of cosines can be used:

${\displaystyle {\begin{aligned}\alpha &=\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}\\[4pt]\beta &=\arccos {\frac {a^{2}+c^{2}-b^{2}}{2ac}}.\end{aligned}}}$

Then angle γ = 180° − α − β.

Some sources recommend to find angle β from the law of sines but (as Note 1 above states) there is a risk of confusing an acute angle value with an obtuse one.

Another method of calculating the angles from known sides is to apply the law of cotangents.

Two sides and the included angle given (SAS)Edit

Two sides and the included angle given

Here the lengths of sides a, b and the angle γ between these sides are known. The third side can be determined from the law of cosines:

${\displaystyle c={\sqrt {a^{2}+b^{2}-2ab\cos \gamma }}.}$

Now we use law of cosines to find the second angle:

${\displaystyle \alpha =\arccos {\frac {b^{2}+c^{2}-a^{2}}{2bc}}.}$

Finally, β = 180° − α − γ.

Two sides and non-included angle given (SSA)Edit

Two sides and a non-included angle given

Two solutions for the triangle

This case is not solvable in all cases; a solution is guaranteed to be unique only if the side length adjacent to the angle is shorter than the other side length. Assume that two sides b, c and the angle β are known. The equation for the angle γ can be implied from the law of sines:

${\displaystyle \sin \gamma ={\frac {c}{b}}\sin \beta .}$

We denote further D = c/b sin β (the equation's right side). There are four possible cases:

1. If D > 1, no such triangle exists because the side b does not reach line BC. For the same reason a solution does not exist if the angle β ≥ 90° and b ≤ c.
2. If D = 1, a unique solution exists: γ = 90°, i.e., the triangle is right-angled.
3. If D < 1 two alternatives are possible.
   1. If b ≥ c, then β ≥ γ (the larger side corresponds to a larger angle). Since no triangle can have two obtuse angles, γ is an acute angle and the solution γ = arcsin D is unique.
   2. If b < c, the angle γ may be acute: γ = arcsin D or obtuse: γ′ = 180° - γ. The figure on right shows the point C, the side b and the angle γ as the first solution, and the point C′, side b′ and the angle γ′ as the second solution.

Once γ is obtained, the third angle α = 180° − β − γ.

The third side can then be found from the law of sines:

${\displaystyle a=b\ {\frac {\sin \alpha }{\sin \beta }}}$

or

${\displaystyle a=c\cos \beta \pm {\sqrt {b^{2}-c^{2}\sin ^{2}\beta }}}$

A side and two adjacent angles given (ASA)Edit

One side and two adjacent angles given

The known characteristics are the side c and the angles α, β. The third angle γ = 180° − α − β.

Two unknown side can be calculated from the law of sines:

${\displaystyle a=c\ {\frac {\sin \alpha }{\sin \gamma }};\quad b=c\ {\frac {\sin \beta }{\sin \gamma }}.}$

or

${\displaystyle a=c{\frac {\sin \alpha }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}}$

${\displaystyle b=c{\frac {\sin \beta }{\sin \alpha \cos \beta +\sin \beta \cos \alpha }}}$