Closing error and graphical adjustment

Closing error (misclosure) is the gap by which a closed traverse fails to close when it is plotted from the observed bearings and lengths. Starting at A, the plotted end falls at A′ instead of A; the distance A′A is the closing error. It arises from errors in the measured angles/bearings, errors in the measured lengths, local attraction, and plotting errors.

The accuracy is judged by the relative error of closure: $\dfrac{\text{closing error}}{\text{perimeter}}$ (acceptable for compass work if about 1/300 or smaller).

Graphical adjustment by Bowditch's rule

Bowditch's rule says that the correction to each station is proportional to the length of traverse up to that station, so that the whole closing error is removed at the last station. It can be applied without calculation:

1. Plot the traverse A-B-C-D-E-A′; the closing error is A′A.
2. Draw a straight base line AA′ equal to the perimeter and mark b, c, d, e at distances AB, AB+BC, AB+BC+CD, ... (in any scale).
3. At the end A′ erect a perpendicular A′X equal to the closing error (to the plan scale) and join A to X.
4. The vertical ordinates bb′, cc′, dd′, ee′ drawn from the base to the line AX are the corrections for stations B, C, D, E. They increase uniformly, and the last ordinate equals the full error.
5. On the plan, from each of B, C, D, E draw a line parallel to the closing line A′A and mark off the corresponding correction towards A. Join the new points in order to get the adjusted closed traverse.

                                      X
                                    . |
                                .     |
                          . e'        |
                    . d'  |           |
              . c'  |     |           |
        . b'  |     |     |           |
 A -----b-----c-----d-----e-----------A'
 (base = perimeter, marks at cumulative distances;
  A'X = closing error, drawn at right angles)

The method is satisfactory only when the error is small (about 1/300 of the perimeter); a larger error means a mistake in the field, and the survey should be repeated.