HOW TO DETERMINE ANGULAR MISCLOSURE AND IT'S ADJUSTMENT, WITH A WORKED EXAMPLE

For a closed traverse, before any coordinate calculations can commence, the whole circle bearing of all the lines have to be calculated.

• The first stage in the calculation process is to check that the observed angles sum to their required values.

The observed angles of a polygon traverse can be either the internal(interior) or external(exterior) angles.

• The angular misclosures are found by comparing the sum of the observed angles with one of the following theoretical values;

                               Sum of internal angles = (2n - 4)Ⅹ 90⁰

                             Or

                               Sum of external angles = (2n - 4)Ⅹ 90⁰   

                               where n is the number of angles measured 

From known points we can determine the initial back bearing and the bearing in the linked traverse can be calculated. To check the observed angles a final forward bearing is computed from known points at the end of the traverse.

• The angular misclosure in a link traverse can be found using this simple formula :                       Sum of angles = (final forward bearing - initial back bearing) + m Ⅹ 180⁰                          

where m is an integer, the value of which depends on the shape of the traverse. 

In most cases, m will be (n - 1), n or (n + 1) where n is the number of angles measured between the initial back bearing and final forward bearing.

     Note: For both types of  traverse care must be taken to ensure that the correct angles have been abstracted and summed, which is in internal or external angles in a polygon traverse and the angles on the same side of a link traverse.

• When the angles have been summed and checked, a very large misclosure probably means that an incorrect angle has been included or one of the angles has been excluded.

ALLOWABLE MISCLOSURE:

   Owing to the effects of occasional miscentring, slight misreading and small bisection errors, a small misclosure will result when the summation check is made.

                       The allowable misclosure E is given by:

                                E'' = 〸or - KSn½

where      K : is a multiplication factor of 1 to 3 depending on weather conditions, number of rounds taken and so on.    

                S : is the smallest reading interval on the theodolite in seconds. example says 60'' , 20'' , 1''.

                n : is the number of angles measured.

       ADJUSTMENT

When the actual misclosure is known and is compared to its allowable value , Two cases may arise.

1. If the misclosure is acceptable ( less than the allowable) it is divided equally between the observed angles. An equal distribution is the only acceptable method since each angle is measured in the same way and there is an equal chance of the misclosure having occured in any of the angles.

  Note: No attempt should be made to distribute the misclosure in proportion to the size of an angle.

      2. If the misclosure is not acceptable ( greater that the allowable) the angles should be remeasured if no gross error can be located in the angle bookings or summation.

                    WORKED EXAMPLE

STATIONS            OBSERVED ANGLES         ADJUSTMENT         ADJUSTED ANGLES

     A                              115⁰  11'   20''                            - 20''                           115⁰  11'   20''

     B                              95⁰    00'   20''                           - 20''                            95⁰    00'   00''

     C                             129⁰   49'   20''                           - 20''                           129⁰   49'   00''

     D                             130⁰   36'   20''                           - 20''                           130⁰   36'   00''

     E                             110⁰   30'   00''                           - 20''                            110⁰   29'   40''

     F                             138⁰   54'   40''                          - 20''                             138⁰   54'   20''

    

SUMS                         720⁰   02'   00''                         -02'   00''                        720⁰   00'  00''

Required sum = [( 2* 6)- 4) * 90⁰                                                               Adjustment per angle

                        = 720⁰  00'  00''                                                                       = -(02'   00'')/ 6

where n here is = 6                                                                                         = - 20''

Misclosure = + 02'  00''

checked