1. Angle Condition

a.Interior Angles

For a closed non-crossing loop traverse, the angle condition is: 

b1  

img1

Eqn (B-1)

Figure B-1
Angle Condtion - Loop
   
 A loop traverse which crosses itself doesn’t have conventional “interior” angles Eqn (B-1) won’t work. Some other angle condition must be established for such cases.    b2
  Figure B-2
Crossling Traverse

 

b. Deflection Angles

If deflection angles are used, the angle condition depends on how many times, if any, the traverse crosses itself. 

 b3    b4
(a) Even crossings   (b) Odd crossings
Figure B-3
Deflection Angles
 img2    img3
Eqn (B-2)   Eqn (B-3)

 

Angle condition equations are exact so the 0°, 180°, and 360° in the equations are also exact.

c. Link Traverse

In order to determine angular misclosure for a closed link traverses, directions must be known at both ends:

b5
Figure B-4
Link Traverse Angle Condition

2. Angular Misclosure

Angular misclosure is the difference between the measured angles' sum on a traverse and the angle condition for the traverse configuration.
The amount of allowable angular misclosure is dependant on the survey purpose. The limits are generally expressed as an Error of a Series:

img4    Eqn (B-4) 
C: Allowable misclosure, seconds
k: expected error in each angle, seconds
n: number of angles

 

Traditional formal standards from the FGCS Standards and Specifications for Geodetic Control Networks are:

Table B-1
Angular Misclosure
Order Class k (sec)
First - 1.7
Second I 3.0
  II 4.5
Third I 10.0
  II 12.0

 

Example

What is the allowable angular misclosure for a five-sided traverse if it is to meet Second Order Class II standards, Figure B-5?

b9
Figure B-5
Allowable Misclosure 

 

From Table 1, k = 4.5". Substituting into Eqn (B-4):

img5

Because this is a five-sided loop traverse, using Eqn (B-1) the angles should sum to:

img6

Therefore, the total should be in the range 539°59'50" to 540°00'10".

 

Informal standards can be created which reflect equipment, experience, conditions, project needs, etc.

Example

A control network is to be established for a construction project. Sight distances will be relatively short (under 300 ft) so measuring angles consistently will be a challenge. Testing under these conditions, our survey crew was able to measure an angle to a consistency of ±0°00'12". For this project, we will use an allowable misclosure of:

img7


3. Angular Misclosure Distribution

Normally, the first step in traverse computations is distribution of the angular misclosure in order to meet the angular condition. This is referred to as angle adjustment. If performing a least squares traverse adjustment, angles are not adjusted since the angular misclosure is part of the overall adjustment.

Because angular misclosure is due to random errors, a distribution method which approximates their behavior would be the best to use. There are a few simplified methods which can be applied, each having advantages and disadvantages, Table B-2: 

Table B-2

Method

Process

Advantage

Disadvantage

Don’t distribute

- -

Simple; Can be replicated.

Unless using a least squares adjustment, this method does not try to reconcile error accumulation.

Arbitrary

Put all error into random angle.

None.

Doesn't model errors well; Difficult to replicate.

Equal distribution

Apply equal correction to each angle. 

Simple; Can be replicated.

Assumes error is same in each angle; Can imply incorrect angle accuracy.

Judgment

Apply different corrections to each angle depending on measurement repetition and conditions.

Relatively simple; Better error modeling.

Requires direct knowledge of measurement conditions.

 

One thing we do not base corrections on is angle size. An angle is the difference between a total station’s initial and final readings; size of the angle has no impact on its error.

Example

Given the following traverse and measured angles:

b6

Point

Angle

A

125°30'20"

B

88°21'31"

C

112°38'35"

D

104°21'40"

E

109°07'41"

sum:

539°59'47"

Figure B-6
Example Angle Adjustment 

 

The angle condition from Eqn (B-1) is

img6

img8

Since the angle total is short of the angle condition, corrections must be added.

Using an equal distribution:

img9

Apply correction:

Point

Angle

corr’n

Corr’d Angle

A

125°30'20"

+0°00'02.6"

125°30'22.6"

B

88°21'31"

+0°00'02.6"

88°21'33.6"

C

112°38'35"

+0°00'02.6"

112°38'37.6"

D

104°21'40"

+0°00'02.6"

104°21'42.6"

E

109°07'41"

+0°00'02.6"

109°07'43.6"

sums:

539°59'47"

+0°00'13.0"

540°00'00.0"

   

check

check

 

While the corrected angles total the angle condition, each is shown to 0.1" which implies a higher accuracy than shown in the original angles. But rounnding each to the nearest second raises the total to 540°00'05".

Using judgment

In this case, we take into account measurement conditions.

Consider point E, which has short sights on both sides. Shorter sights generally have higher pointing errors (given same target sizes).

Also, the field crew reported larger repeated direct and reverse angle spreads at point C.

The equal distribution’s correction was +0°00'02.6" per angle. We can drop that to +0°00'02" per angle for our three strongest angles (+0°00'06" total) and divide the remaining 0°00'07" among the two weaker ones.

 

Point

Angle

corr’n

Corr’d Angle

A

125°30'20"

+0°00'02"

125°30'22"

B

88°21'31"

+0°00'02"

88°21'33"

C

112°38'35"

+0°00'04"

112°38'39"

D

104°21'40"

+0°00'02"

104°21'42"

E

109°07'41"

+0°00'03"

109°07'44"

sums:

539°59'47"

+0°00'13"

540°00'00"

 

 

check

check

 

This takes into account individual conditions. The resulting angles do not imply accuracy beyond the original measurements.