Saturday, March 31, 2018

What are Mistakes,Systematic error and Accidental Errors, in Surveying

This post contains mostly the theory of the Errors, as we define them in Surveying. 

Error:- The difference between the observed value and the true value is known as error.
There are three types of errors which occurs while we do the surveying:
  1. Mistakes: These are the errors which occur due to the inexperience, inattention, carelessness or due to lack of judgement or poor judgement. If mistakes are not found then they may affect the result to a great extent.
  2. Systematic errors: These are the errors which follow a system when they occur, and they have the same nature whenever they occur. They can be eliminated by testing the instruments before they are used or by applied the necessary correction, by using the mathematical formulae after the error is known.
  3. Accidental errors: These are the kind of errors which occur accidentally and can of any nature positive or negative. These are the errors which are byond the human control and can not be calculated to their true value, but only we can apply the theory of the probability to calculate them.
Weight of an observation: Weight of an observation is its relative importance to the other observations taken under the identical conditions.

Mean value: Mean value of a set of observation is the arithmetic or the weighted arithmetic mean of the observations.

Probable Error of a single observation:
Probable error of a single observation in a set of observation is derived using the formula,
Es = +-0.6745(v2/ n-1)
where, v= difference between the observation and the mean value of the set of observation.
And n= numbers of observations in the given set.

Probable error of the mean:
The probable value of the error in a mean of the set of observation can be determined using the following formula = Es/√n

Principle of least squares:
Most probable value of a given quantity from the given available set of observation is the one for which the sum of the squares of the residual errors is a minimum.
Alternatively, the most probable values of the errors in the given set of observations of equal weight are those for which the sum of their squares is a minimum.

It can be proved that the mean value is the true value in case the numbers of observations are very large in numbers.
The sum of the squares of the residuals found by the arithmetic mean value is a minimum. This is thus the fundamental law of least squares. 

Adjustments of the errors in the triangulation:

The triangulation errors are adjusted in three different ways or steps:
(a) Adjustments of the single angle error (b) Adjustments of the station observation
(c) Adjustments of the figures

(a) Adjustments of the single angle error: When an angle is measured a numbers of times, then the most probable value of the angle is the mean value of the observed values. If all the values observed are of the same weight then the most probable value is the simple arithmetic mean, but if the observations taken are of different weight then the most probable value is the weighted arithmetic mean.

(b)Station adjustments:
When there are numbers of observations taken at the same station, then the condition that the sum of all the angles should be equal to 360 degrees must be satisfied, if not then the difference from 360 is the error of that station. Now there arise two more cases:
(i) When all the angles measured have the same weight: In this case the error is distributed equally among all the angles. 
(ii) When the angles are of different weight: In this case the error is distributed in the proportion of inverse of the weight of the different angles.  
(iii) When the angles and some other combined observations are taken: In such cases when combined observations are also taken along with the observation of the single angles, we have to make use of the normal equation, to find out the errors. 
There is another method known as the method of difference, which can be used in more simple way to get the errors, because the method of normal equations is more laborious. 

(c) Adjustments of the figures: 
There are different conditions which can be opted to calculate the errors of the different angles. In a triangular figure, the sum of three angles is always equal to 360 degrees. 
Similarly there can be other conditions which can opted to find out the condition equations for other figures like quadrilateral, or central figures. 
For a closed traverse, sum of internal angles is (2n-4)*90 degrees, where n is the no. of sides, or total no. of angles. 

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