Traverse Computations

1
Traverse Computations

• Purpose
• Approaches to adjustment

2
Compass Rule Adjustment

• Procedure

3
Example Traverse
B
38.576 m
110 05' 00"
C
30.141 m
111 25' 30"
25.605 m
65 33' 30"
A
72 54' 30"
58.437 m
D
4
Balancing Angles

• Purpose
• Procedure

5
Methods for Adjusting Angles

• Several possibilities
• Selection of method

6
Equal Corrections

• Issue
• Distribute closure

7
Equal Corrections to Angles

• Advantages

8
Example Traverse Balanced Angles
B
38.576 m
110 05' 30"
C
30.141 m
111 26' 00"
25.605 m
65 33' 30"
A
72 55' 00"
58.437 m
D
9
Directions of Lines

• Use adjusted angles to compute line direction
• Azimuths or bearings?

10
Computing Directions
B
direction?
110 05' 30"
C
30 15' 30"
A
11
Compute Directions Clockwise Around Traverse
B
100 10' 00"
110 05' 30"
C
30 15' 30"
Assume or find azimuth of AB 30 15' 30"
Compute azimuth of BA (azimuth of AB ? 180)
210 15' 30" Subtract angle at B 210 15' 30"
- 110 05' 30" 100 10' 00"
A
12
Directions of Traverse Lines
13
Directions of Traverse Lines
B
100 10' 00"
C
30 15' 30"
168 44' 00"
A
275 49' 00"
D
14
Departures and Latitudes

• Departures (deps)
• Latitudes (lats)

15
Computing Departures Latitudes

• Compute by Dep L sin ? Lat L cos ?
• Where ? azimuth L length of line

North (Y)
G
?
Lat. FG L cos?
L
F
East (X)
Dep. FG L sin?
16
Example Traverse
B
100 10' 00"
38.576 m
C
30 15' 30"
30.141 m
168 44' 00"
25.605 m
A
58.437 m
275 49' 00"
D
17
Example Traverse
Departure Length x sin (azimuth)e.g. Dep. AB
30.141 m x sin(30 15' 30") Latitude Length x
cos (azimuth) e.g. Lat. AB 30.141 m x cos (30
15' 30")
18
Closure in Deps and Lats

• For mathematically closed traverse
• Geometrically closed traverse
• Closure difference between known/computed
  position
• Linear error of closure (LEC)
• Relative error of closure (REC)

19
Linear Error of Closure
?dep 0.0248 m
A'
?lat 0.0363 m
0.044 m
A
20
Traverse Adjustment

• Goal
• Some methods
• Arbitrary method
• Compass (Bowditch) rule
• Least squares adjustment

21
Readings

• Chapter 10 sections 10.1 10.8

22
Compass Rule Adjustment

• Application
• Works for traverses with limited number of lines

23
Compass Rule

• Proportion is rearranged for computational
  efficiency

24
Compass Rule Balance Departures
Corrected departure Departure Departure
correctione.g. Corrected departure AB 15.1880
(-0.0049) 15.183 m
25
Compass Rule Balance Latitudes
Corrected latitude Latitude Latitude
correctione.g. Corrected latitude AB 26.0347
(-0.0071) 26.028 m
26
Computation of Coordinates

• General
• Application

27
Calculate Coordinates
28
Adjusted Azimuths and Lengths
North (Y)
G
?
Lat. FG
L
F
L length of line FG
East (X)
Dep. FG
? azimuth of line FG
29
Calculating Azimuths and Lengths
L length of line JK
North (Y)
? azimuth of line JK
Dep. JK
East (X)
J
?
L
Lat. JK
K
30
Compute Adjusted Azimuths/Lengths
31
Least Squares Adjustment

• Theory
• Application
• Issue

32
Understanding LS

• Advantages of LS approach
• Assumptions

33
Understanding LS

• Generally make redundant measurements
• Residual
• Difference between observation and MPV
• e.g. ?i MPV obsi
• Premise of LS

34
Example

• Measure distance five times
• 100.04, 100.15, 99.97, 99.94, 100.06
• Every observation has associated residual
• M 100.04 ?1 ? ?1 M 100.04
• M 100.15 ?2 ? ?2 M 100.15
• M 99.97 ?3 ? ?3 M 99.97
• M 99.94 ?4 ? ?4 M 99.94
• M 100.06 ?5 ? ?5 M 100.06
• LS selects M to minimize sum of squared residuals

35
Example cont.

• Want to minimize sum of residuals

• Recall from calculus
• Function minimum occurs when derivative is zero
• ?Take derivate with respect to M and equate to
  zero

36
LS Adjustment of Traverse

• Components
• Measure angles and distances
• Derive MPV of point coordinates
• Observation equations
• Express observed measurements in terms of
  coordinates

• Issue
• Equations are non-linear
• Normalized through Taylor expansion

37
Horizontal Traverse LS Adjustment

• Minimal input

38
Estimating Horizontal Coordinates

• Approach

39
Input File for LS Adjustment
40
Output File for LS Adjustment

• Contents of output file
• Summary of input data
• Adjusted stations
• Adjusted distances and residuals
• Adjusted angles and residuals
• Adjusted azimuths and residuals
• Associated statistics
• Quality of adjustment
• Size of all residuals
• Consider SD of unit weight

41
Output From LS Adjustment
42
Readings

• Chapter 10 sections 10.11, 10.16 10.17
• Chapter 15 sections 15.1 15.3