A. Coordinates Galore

Most software compute and carry coordinates through each step of the traverse computations. Until final traverse adjustment, the coordinates go thru a series of values. For example, in a traditional non-least squares approach coordinates would be computed:

  • From raw measurements and directions using unadjusted angles, then,
  • After angles are adjusted, and,
  • After the traverse is adjusted - these are the final coordinate values.

While coordinates are not needed for manual computations, they can be included. If so they are usually computed after angles are balanced: we will refer to these as preliminary coordinates


B. Adjustment Process

The traverse misclosure is determined by comparing the preliminary coordinates of the end point to its known starting coordinates:

Loop traverse, Figure I-1:

 img200

Figure I-1
Loop Traverse Closure

 

On a loop traverse, the coordinates of the first point can be known or assumed.

Link traverse, Figure I-2.

The coordinates of the traverse endpoints must be known in the same system.

In this case E and H are known.

 

img204

Figure I-2
Link Traverse Closure

 

When the traverse is adjusted, corrections are applied directly to the preliminary coordinates (J') to obtain final coordinates (J), Figure I-3 and Equation I-1.

Figure I-3
Applying Corrections

 

      Eqn (I-1)

 

The line correction affects the line's endpoint, that is, the position of J changes relative to the position of I by the latitude and departure corrections of the line IJ.

Because the coordinates of J change, the coordinates of the next point, K, must change by the same amount plus the correction of the line JK, Figure I-4:

Figure I-4
Cumulative Corrections

 

The position shift from K' to K'' is the same as the shifts from J' to J.

This pattern of accumulating corrections continues until the final point where the adjusted coordinates should equal the known values.

Using the Compass Rule:

      Eqn (I-2)

 

To compute coordinate corrections, the equations are modified to: 

      Eqn (I-3)

 

where ΔNi-1 and ΔEi-1 in each are the cumulative previous corrections.


C. Examples

The traverse examples here are the same ones used as previous computation examples so we can check these against earlier results.

1. Bearing Traverse

 

Figure I-5
Bearing Traverse Example

 

Previously computed unadjusted latitudes and departures are:

Line Bearing Length (ft) Lat (ft) Dep (ft)
AB S 68°05'35"W 472.68 -176.357 -438.548
BC N 19°46'00"W 216.13 +203.395 -73.093
CD N 45°55'20"E 276.52 +192.357 +198.651
DA S 54°59'15"E 382.24 -219.312 +313.065
 

sum:

1347.57 +0.083 +0.075

 

(a) Compute preliminary coordinates from the raw latitudes and departures:

(b) Determine latitude and departure closure errors:

Lat err = NA' - NA = 500.083' - 500.000' = +0.083'

Dep err = EA' - EA = 2000.075' - 2000.000' = +0.075'

Note that these match the latitude and departure column sums above.

(c) Compute and apply corrections to the coordinates using the Compass Rule, Equation (I-3).

Line AB

Because point A is a control point, its coordinates are not modified. The corrections are applied to the line's endpoint, point B.

Line BC

Remember to include the corrections for the previous line.

Line CD

Line DA

The corrected preliminary coordinates match the beginnnig coordinates of point A.

These coordinates also match those computed in the Coordinates section.

2. Link Traverse

 

Figure I-6
Link Traverse Example

 

Previously computed unadjusted latitudes and departures are:

Line

Direction

Length

Lat

Dep

QR

S 56°23'38"E

398.75'

-220.700'

+332.104'

RS

S 75°17'42"W

422.89'

-107.347'

-409.038

ST

N 43°05'47"E

604.49'

+441.402'

+413.004'

 

sums:

1426.13'

+113.355'

+336.070

 

(a) Compute preliminary coordinates:

Point

North (ft)

 

East (ft)   

Q

2600.480

 

1391.670   

LatQR

-220.700

DepQR

+332.104   

R

2379.780

 

1723.774   

LatRS

-107.347

DepRS

-409.038   

S

2272.433

 

1314.736   

LatST

+441.402

DepST

+413.004   

T

2713.835

 

1727.740   

 

Lat err = 2713.835' - 2713.780' = +0.555'

Dep err = 1727.740'-1727.810' = -0.070'

(c) Compute and apply corrections to the coordinates using the Compass Rule, Equation (I-3).

Computations for each line are not shown, but their results are tabulated below.

Point

N' (ft)  

ΔN (ft) 

N (ft)  

E' (ft)  

ΔE (ft) 

E (ft)  

S

2379.780

-0.015

2379.765

1723.774

+0.020

1723.794  

T

2272.433

-0.031

2272.402

1314.736

+0.041

1314.777  

Q

2713.835

-0.054

2713.781

1727.740

+0.071

1727.811  

     

check

   

check