Surveying (Part 15)
The relation between the observed quantity and its numerical values is known as observation equation.
Normal Equation The equation which is obtained by multiplying each equation by the coefficient of the unknown whose normal equation is to be found and by adding the equation thus formed is known as normal equation. The number of normal equation is always the same as the number of the unknowns.
Conditioned equation The equation which express the relation existing between the several dependent quantities is known as conditioned equation.
Laws of weight The methods of Least squares for errors adopts the following laws of weights
1. The weight of the arithmetic mean of a number of observation of unit weight is equal to the no. of the observations.
2. The weight of the weighted arithmetic mean of a numbers of observation is equal to the sum of the individual weight of observations In this method, following steps are involved
(1) Assume suitable correction for each observed quantities e1, e2, e3........en
(2) Enter all the conditions equation i.e.e1+ e2+e3+e4 = E e2 + e3 = F
(3) Add the equation of condition as suggestd by the theory of least squares
i.e. 22 w1 e1 + w 2 e2 + ..........= a minimum
(4) Differentiate each conditioned eqation and the equation of least squares separately.
(5) Multiply each differentiated equation of conditions by correlates – a1, a2, a3,......ln etc. and add the result to the differentiated equation of the least squares.
(6) Obtain the co-efficients of de1 , de2 , ........den and equated each to zero to get the values of e1, e2, e3.......
(7) Substitute the values of e1, e2, e3 .... etc in the condition equation and solve the simultaneous equation thus formed to obtain the values of correlates.
(8) Knowing the values of correlates and the weight , the values of e1, e2, e3.....etc can be calculated
To form the normal equations, “Multiplied each observation equation by the product of the algebraic coefficient of that unknown quantity in the equation and the weight of that observation and add the equation thus formed”
(a) It may be left of C
(b) It may lie right of C
(c) It may be lie below C
(d) It may lie above C
(7) Computation of Lalitude and Departures.
Let the bearing of side CA is known \ Bearing of side AC = bearing of CA – 180 Bearing of side AB = Bearing of AC + ÐCAB Bearing of BC = Bearing of BA + ÐABC Knowing the co- ordinate of A, the co-ordinate of B and C be computed as under Latitude of C = AC × cosine of the R.B of the (Northing or Southing) Departure of C = AC × sine of the R.B of AC (Easting or Westing)
(8) The weight of the quotient of any quantity divided by a constant is equal to the weight of the quantity multiplied by the square of that constant.
(9) The weight of an equation by multiplying the weight of the given equation is equal to the reciprocal of the weight of the given equation.
(10) The weight of an equation remain unchanged if all the signs of the equation are changed or if the equation is added to an subtracted from a constant. The arithmatic mean is the most probable value of number of observations of equal wts.
Most probable values of directly observed quantities.
(i) Equal weights. The most probable of the observed quantity is equal to the arithmetic mean of the observed values of the quantity.
where X1, X2, X3 ..... are the observed values of a quantity
(ii) Different weights. The most probable value of the observed quantity is equal to the weighted arithmetic mean of the observed values of the quantity.
Residual errors The difference between any measured quantity and the most probable value of that quantity is known of residual error.
Normal tension For normal tension correction due to pull and correction due to sag is neutralises.
normal tension P can be found
Measurement of horizontal angles The horizontal angle of a triangulation system can be observed by the following methods
(1) The Repetition method
(2) The Reiteration method
THEORY OF ERRORS AND ADJUSTMENT
Broadly. We may divide various errors into the following three main categories.
(1) Mistakes. Arise due to carelessness in attention and confusion by inexperienced workers.
(2) Systemtic errors. These errors which always follow some definite mathematical and physical laws are termed as systematic errors. e.g., wrong length of a chain, atmospheric refraction
(3) Accidental or random errors or compensating erros. Those errors which re-main even after eliminating mistakes and systematic errors are caused due to a combination of various reasons beyond the control of the observers are termed as accidental errors. Random errors are generally small and can never be avoided.
Definition True value. The value of a quantity which is absolutely free from all types of errors is termed as the value.
Which is never known.
Observed value. The value of a quantity which is obtained after correcting it for known errors is known as observed value. Observed quantity may further classified in
(a) independent quantity e.g., R.L of bench mark
(b) conditioned quantity True error. True value- Observed value.
Most probable value. It is that value of a quantity which has more chances of being true.
Most probable error. It may be defined as the quantity which is subtracted from or added to the most probable value.
Satellite computation. Observation are taken at the satellite station
According to the position of satelite station (S) with regard to the side of AC and BC, the following four cases may arises
2. The triangulation stations which are used only to provide additional rays to intersect points are known as subsidiary stations.
3. The stations which are selected close to the main triangular stations to avoid inter vening obstructor, are known as satellite stations or eccentric or false stations.
4. The stations at which no obstructions are made but the angles of these stations are used for the continuity of a triangular series, are known as pivot stations.
(1) Checking the means of the observed angles
(2) Checking the triangular errors
(3) Checking the total round of each station
(4) Computation of the length of the base line
(5) Computation of the sides of the main triangle
(6) Satellite computation if any.
(7) Sag correction =
where, W = Total weight of the tape L = Horozontal distance between
[Since W= wL]
(8) Reduction ot M.S.L.
The difference in the length of the measured line and its equivalent ength at sealevel, is known as error due to reduction to M.S.L.
L = L – L'
(4) Correction for Tension :
where,P0 = standard pull
P = pull applied during measurement
If applied pull in more, tension correction is+ve, and if it is less, tension correction is ‘–’ ve
II. When the observation is made on the base line
Ð SCO =180 - (q -b)
D PCO 180 -
DCPO 180 -
The phase correction is applied algebraically to the observed angle according to the relative positions of the sun and the signal, i.e., it is the +ve and zero.