COMBINATION OF ERRORS
Much data in surveying is obtained indirectly from various combinations of observed data, for instance the coordinates of the ends of a line are a function of its length and bearing. As each measurement= contains an error, it is necessary to consider the combined effect of these errors on the derived
The general procedure is to differentiate with respect to each of the observed quantities in turn and sum them to obtain their total effect. Thus if a = f (x, y, z, . . .), and each independent variable changes by a small amount (an error) δx, δy, δz, . . . , then a will change by a small amount equal to δa, obtained from the following expression:
in which ∂a/∂x is the partial derivative of a with respect to x, etc.
Consider now a set of measurements and let the residuals δxi, δyi, and δzi, be written as xi, yi, and zi and the error in the derived quantity δaI is written as ai:
Now squaring both sides gives
In the above process many of the square and cross-multiplied terms have been omitted for simplicity. Summing the results gives
As the measured quantities may be considered independent and uncorrelated, the cross-products tend to zero and may be ignored.
Now dividing throughout by (n − 1):
The sum of the residuals squared divided by (n − 1), is in effect the variance σ2, and therefore'
which is the general equation for the variance of any function. This equation is very important and is used extensively in surveying for error analysis, as illustrated in the following examples.