The Statistics Of
    Measurement
 Types Of Errors
 Random Errors
 Normal Distribution
 Standard Errors Of
    Derived Quantities
 Propagation Of Small
    Systematic Errors

Standard Errors of Derived Quantities - Propagation of Variances

If we have a known precision of reading of a level staff, and we perform a 5km level traverse, what is the likely precision of the Reduced Levels? This question is common to many surveying and measurement tasks, what will be the derived precision of a quantity based on the measurements we take. One way this is determined is to use the rule for the propagation of variances.

If X is a function of ‘n’ independent random variables x_i with an expected mean of m and standard deviation of s. The data needs to be free of correlations, that is that a change in one observation does not affect any other observation.

This is estimated by the use of 's' in the formula instead of the unknown s. This is an important formula for geomatic engineers.

Now, getting back to our levelling question:

So, in order to determine the final precision of a level traverse, we use the rule for the propagation of variances:

and if all staff readings have the same precision,

root of the number of set-ups, which is the criterion used in the practical exercises. As the number of set-ups is proportional to the distance, the precision of the RL is also proportional to the distance, which is how the Survey Coordination Regulations express accuracies for levelling traverses.

For another example, consider a change in height determined by measuring a radiation with a total station:

From the diagram: DH = HI + D Sin q - HT, so....

s²_DH = s²_DI + Sin²qs²_D + D² Cos²qs²_q + s²_HT

If we considered the precision of say the angle reading only, then the precision of the change in height will be:

s²_DH = D²Cos²qs²_q

s_DH = D Cos qs²_q

So, for a distance of 500m, a slope of 8°00’ and a precision of angle measurement of 10", the precision will be:

Taking another example, what is the derived precision of the coordinates of a point obtained from the measurement of a radiation with a total station?

Data: coordinates of A and B

Measurements: length AC (l), variance s²_l

angle CAB, a, variance s²_l

The quantities that are to be derived are the bearing between A and C, and then the coordinates of C.

E_C = E_A + l x Sinq_AC = E_A + l x Sin(q_AB - a)

N_C = N_A + l x Cosq_AC = N_A + l x Cos(q_AB - a)

So then the variance of the computed coordinates is:

s2E = (	DEC	)2s2l + (	DEC	)2s2a = Sin2(qAB - a)s2l + l2Cos2(qAB - a)s2a
	dl		da
similarly, s2N = Cos2(qAB - a)s2l + l2Sin2(qAB - a)>s2a

For example, let's compute the precisions of some coordinates given a distance and a series of bearings:

Distance = 100m, precision of distance measurement = +0.01m

Bearing = 0°, 30°, 60°, 90, precision of angle measurement = +05"

It can be seen that the precision of the coordinate varies depending on the bearing of the line, whereas the precisions of the measurements are constant.

The rule of propagation of variances can be applied to most measurement applications, enabling the precision of even complex derivations to be determined or predicted.