CHAPTER 16 ADJUSTMENT OF HORIZONTAL SURVEYS: TRAVERSES AND HORIZONTAL NETWORKS
16.1 INTRODUCTION TO TRAVERSE ADJUSTMENTS
Of the many methods that exist for traverse adjustment, the characteristic that distinguishes the method of least squares from other methods is that distance, angle, and direction observations are adjusted simultaneously. Furthermore, the adjusted observations not only satisfy all geometrical conditions for the traverse, but they also provide the most probable values for the given set of data. Additionally, the observations can be weighted rigorously based on their estimated errors and adjusted accordingly. Given these facts, together with the computational power now provided by computers, it is hard to justify not using least squares for all traverse adjustment work.
In this chapter, we describe methods for making traverse adjustments using the least squares method. As was the case in triangulation adjustments, traverses can be adjusted by least squares using either observation equations or conditional equations. Again, because of the relative ease with which the equations can be written and solved, the parametric observation equation approach will be discussed.
16.2 OBSERVATION EQUATIONS
When adjusting a traverse using parametric equations, an observation equation is written for each distance, direction, or angle. The necessary linearized observation equations developed previously are recalled in the following equations.
The reader should refer to Chapters 14 and 15 to review the specific notation for these equations. As demonstrated with the examples that follow, the azimuth equation may or may not be used in traverse adjustments.
16.3 REDUNDANT EQUATIONS
As noted earlier, one observation equation can be written for each observed angle, distance, or direction in a closed traverse. Thus, if there are n sides in the traverse, there are n distances and n + 1 angles, assuming that one angle exists for orientation of the traverse. For example, each closed traverse in Figure 16.1 has four sides, four distances, and five angles. Each traverse also has three points whose positions are unknown, and each point introduces two unknown coordinates into the solution. Thus, there are a maximum of 2(n − 1) unknowns for any simple closed traverse. From the foregoing, no matter the number of sides, there will always be a minimum of r = (n + n + 1) − 2(n − 1) = 3 redundant equations for any simple closed traverse. That is, every simple closed traverse that is fixed both positionally and rotationally in space has a minimum of three redundant equations.
16.4 NUMERICAL EXAMPLE
16.5 MINIMUM AMOUNT OF CONTROL
All adjustments require some form of control and failure to supply a sufficient amount will result in an indeterminate solution. A traverse requires a minimum of one control station to fix it in position and one line of known direction to fix it in angular orientation. When a traverse has the minimum amount of control, it is said to be minimally constrained. It is not possible to adjust a traverse without this minimum. If minimal constraint is not available, necessary control values can be assumed and the computational process carried out in arbitrary space. This enables the observed data to be tested for blunders and errors. In Chapter 21, we discuss minimally constrained adjustments.
A free network adjustment involves using a pseudo inverse to solve systems that have less than the minimum amount of control. This material is beyond the scope of this book. Readers interested in this subject should consult Bjerhammar (1973) or White (1987) in the bibliography at the end of the book.
16.6 ADJUSTMENT OF NETWORKS
With the introduction of the EDM instrument, and particularly the total station, the speed and reliability of making angle and distance observations have increased greatly. This has led to observational systems that do not conform to the basic systems of trilateration, triangulation, or traverse. For example, it is common to collect more than the minimum observations at a station during a horizontal control survey. This creates what is called a complex network, referred to more commonly as a network. The least squares solution of a network is similar to that of a traverse. That is, observation equations are written for each observation using the prototype equations given in Section 16.2. Coordinate corrections are found using Equation (11.39) and a posteriori error analysis is carried out.
16.7 χ2 TEST: GOODNESS OF FIT
At the completion of a least-squares adjustment, the significance of the computed reference variance, can be checked statistically. This check is often referred to as a goodness-of-fit test since the computation of is based on ∑v2. That is, as the residuals become larger, so will the reference variance computed, and thus the model computed deviates more from the observed values. However, the size of the residuals is not the only contributing factor to the size of the reference variance in a weighted adjustment. The stochastic model also plays a role in the size of this value. Thus, when the χ2 test indicates that the null hypothesis should be rejected, it may be due to a blunder in the data or an incorrect decision by the operator in selecting the stochastic model for the adjustment. In Chapter 21, these matters are discussed in greater detail. For now, the reference variance of the adjustment of Example 16.2 will be checked.
In Example 16.2, there are 13 degrees of freedom and the computed reference variance, , is 2.2. In Chapter 10, it was shown that the a priori value for the reference variance was 1. A check can be made to compare the computed value for the reference variance against its a priori value using a two-tailed χ2 test. For this adjustment, a significance level of 0.01 was selected. The procedures for doing the test were outlined in Section 4.10, and the results for this example are shown in Table 16.4. Since α/2 is 0.005 and the adjustment had 13 redundant observations, the critical value from the table is 29.82. Similarly its lower-tailed value is 3.57. It can be seen that the computed χ2 value is less than the upper-tailed tabular value and is greater than the lower-tailed tabular value. Thus, the test fails to reject the null hypothesis, H0. As explained in Step 6 of Section 16.4, the a priori value of 1 for can and should be used when computing the standard deviations for the station coordinates and observations since the computed value for the reference variance is only an estimate of the a priori value.
Note: For problems below requiring least squares adjustment, if a computer program is not distinctly specified for use in the problem, it is expected that the least squares method will be solved using a spreadsheet and the program MATRIX, which is included on the book's companion website. Partial answers to problems marked with an asterisk are given in Appendix H.
16.1 Using the following control, adjust the data given for the Figure P16.1. The control station coordinates in units of feet are
Distance Observations
Angle Observations
From
To
Length (ft)
S (ft)
BS
Occ
FS
Angle
S (″)
A
B
2651.08
±0.017
MK1
A
B
285°36′46″
±5.6
B
C
2312.46
±0.017
A
B
C
246°40′29″
±3.2
B
C
MK2
147°41′01″
±3.8
(a) What is the standard deviation of unit weight, S0?
*(b) List the adjusted coordinates of station B and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.2 Adjust by the method of least squares the closed traverse in Figure P16.2. The data are given below.
Observed Angles
Observed Distances
Angle
Value
S (″)
Course
Distance (m)
S (m)
XAB
115°39′54″
±4.4
AB
772.021
±0.015
BAC
22°54′44″
±4.3
BC
705.054
±0.015
CBA
131°51′28″
±5.1
CA
1348.922
±0.016
ACB
25°13′43″
±4.5
Control Stations
Unknown Stations
Station
X (m)
Y (m)
Station
X (m)
Y (m)
X
5581.734
7751.476
B
7345.173
7123.938
A
6634.098
6823.278
C
7983.019
6823.495
*(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.3 Adjust the network shown in Figure P16.3 by the method of least squares. The data are listed as follows:
Control Station
Unknown Stations
Station
X (m)
Y (m)
Station
X (m)
Y (m)
A
1776.596
2162.848
B
1777.144
2457.553
C
1836.902
2533.979
D
1883.089
2200.879
Distance Observations
Angle Observations
Course
Distance (m)
S (m)
Angle
Value
S (″)
AB
294.704
0.0046
CAD
61°07′01″
6.6
BC
97.013
0.0045
BAC
9°07′15″
4.0
CD
336.287
0.0046
CBA
142°05′01″
7.6
DA
113.071
0.0045
ACB
28°47′41″
7.3
AC
376.005
0.0046
DCA
17°07′27″
3.9
ADC
101°45′39″
6.7
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
The observed azimuth of line AB is 0°06′20.5″ ± 1.4″.
16.4 Perform a weighted least squares adjustment using the following distance observations and the angle observations given in Problem 15.3.
From
To
Distance (m)
S
R
U
1188.641
±0.015
S
U
923.222
±0.015
T
U
1295.967
±0.015
(a) What is the reference standard deviation, S0?
*(b) List the adjusted coordinates of the unknown station and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.5 Perform a weighted least squares adjustment using the following distance observations and the angle observations given in Problem 15.6.
Occupied
Sighted
Distance (ft)
S (ft)
P
U
1596.91
±0.016
Q
U
1305.75
±0.016
R
U
1446.62
±0.016
S
U
1277.35
±0.015
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown station and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.6 Perform a weighted least squares adjustment using the following distance observations and the angle observations given in Problem 15.8.
From
To
Distance (ft)
S (ft)
A
B
2925.44
0.015
B
C
2179.01
0.014
C
D
2991.34
0.015
A
C
2740.88
0.015
B
D
4455.26
0.018
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown station and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.7 Using the program ADJUST, do a weighted least squares adjustment using the following distance observations and the angle observations given in Problem 15.9.
From
To
Distance (m)
S (m)
A
B
3111.287
0.010
B
C
2908.209
0.010
C
D
3158.771
0.010
D
A
3260.371
0.011
A
C
4608.981
0.015
B
D
4168.004
0.013
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.8 Repeat Problem 16.7 using the additional data in Problem 15.10.
16.9 Using the program ADJUST, do a weighted least squares adjustment using the following distance observations and the angle observations given in Problem 15.11.
From
To
Distance (ft)
S (ft)
From
To
Distance (ft)
S (ft)
A
D
785.93
0.015
D
E
629.83
0.015
A
C
680.19
0.015
D
F
647.82
0.015
B
C
791.97
0.015
E
F
500.52
0.015
B
D
582.87
0.015
E
G
911.95
0.015
C
D
470.77
0.015
E
H
1053.53
0.015
C
E
342.61
0.015
F
G
764.33
0.015
C
F
720.55
0.015
F
H
664.87
0.015
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.10 Using the Program ADJUST do a weighted least squares adjustment using the following distance observations and the angle observations given in Problem 15.14.
From
To
Distance (ft)
S (ft)
From
To
Distance (ft)
S (ft)
A
E
2377.65
0.014
C
F
1859.84
0.013
A
B
2042.50
0.014
F
J
2150.44
0.014
B
E
2249.55
0.014
I
J
1942.84
0.014
E
H
2071.74
0.014
J
G
1956.66
0.014
E
I
2185.06
0.014
J
K
1845.58
0.013
E
F
2112.41
0.014
G
K
2317.92
0.014
H
I
1957.96
0.014
F
G
1957.04
0.014
F
I
1868.89
0.013
C
G
1859.82
0.013
B
F
1942.92
0.014
D
G
2209.38
0.014
B
C
1668.81
0.013
C
D
2280.09
0.014
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown station and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.11 Using the distances from Problems 14.3 and 14.4, and the following angles perform a weighted least squares adjustment.
BS
Occupied
FS
Angle
S (″)
B
A
C
59°59′58″
±2.9
C
A
D
59°59′56″
±2.5
C
B
D
43°03′12″
±2.3
C
B
A
79°06′29″
±3.0
D
C
A
58°19′07″
±2.4
A
C
B
40°53′38″
±2.5
A
D
B
23°56′47″
±2.2
B
D
C
37°44′05″
±2.2
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.12 Using the distances from Problem 14.10 and the angles listed below do a weighted least squares adjustment.
BS
Occupied
FS
Angle
S (″)
E
A
B
89°32′10″
±2.0
A
B
E
38°19′46″
±2.0
E
B
D
54°04′30″
±2.0
E
B
C
109°24′50″
±2.1
B
C
D
73°11′00″
±2.1
C
D
B
51°28′34″
±2.1
B
D
E
90°34′25″
±2.1
D
E
B
35°21′06″
±2.0
B
E
A
52°08′11″
±2.0
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) List the inverted normal matrix used in the last iteration.
(e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.13 Do a weighted least squares adjustment using the following.
Control Stations
Unknown Stations
Station
E (m)
N (m)
Station
E (m)
N (m)
A
834.133
666.605
B
1608.340
909.299
C
1624.576
1566.347
D
536.804
1622.029
E
1156.384
1276.568
Distance Observations
Angle Observations
Course
Distance (m)
S (m)
Angle
Value
S (″)
AB
811.353
±0.005
DAE
45°08′06″
±2.4
BC
657.246
±0.005
EAB
44°44′51″
±2.4
CD
1089.197
±0.006
ABE
56°30′13″
±2.5
DA
1000.609
±0.005
EBC
52°19′01″
±2.6
EA
689.862
±0.005
BCE
56°49′46″
±2.6
EB
582.365
±0.005
ECD
34°41′07″
±2.5
EC
550.605
±0.005
CDE
26°12′46″
±2.4
ED
709.380
±0.005
EDA
43°34′13″
±2.4
BEA
78°45′04″
±2.6
CEB
70°51′07″
±2.6
DEC
119°06′07″
±2.6
AED
91°17′42″
±2.5
Azimuth Observation
Course
Value
S
AB
72°35′43″
2.3″
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.14 Using ADJUST, do a weighted least squares adjustment using the following data.
Control Stations
Unknown Stations
Station
X (ft)
Y (ft)
Station
X (ft)
Y (ft)
A
51,020.44
50,977.05
B
55,949.19
50,943.17
D
60,369.82
54,974.24
C
60,268.19
51,383.54
E
54,814.39
54,483.06
F
49,885.65
54,533.87
Distance Observations
Angle Observations
Course
Length (ft)
S (ft)
Angle
Value
S (″)
AB
4928.85
±0.019
FAE
64°57′13″
±1.1
BC
4341.36
±0.017
EAB
43°08′06″
±1.0
CD
3592.15
±0.016
ABE
71°49′53″
±1.1
DE
5577.10
±0.020
EBC
101°57′08″
±1.1
EC
6273.01
±0.022
BCE
35°25′56″
±1.0
EB
3717.32
±0.016
ECD
62°00′38″
±1.1
EA
5165.90
±0.019
CDE
83°20′34″
±1.1
EF
4929.11
±0.019
DEC
34°39′45″
±1.0
FA
3733.46
±0.016
CEB
42°36′54″
±1.1
BEA
65°02′09″
±1.1
AEF
43°19′54″
±1.0
EFA
71°42′50″
±1.1
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.15 Repeat Problem 16.14 using the following additional data.
Angle
Value
S
BAF
251°54′42″
1.1″
CBA
186°12′54″
1.1″
DCB
262°33′26″
1.1″
EDC
276°40′27″
1.1″
FED
174°21′26″
1.0″
AFE
288°17′07″
1.1″
16.16 Using ADJUST, do a weighted least squares adjustment using the following data.
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
16.17 Using ADJUST, do a weighted least squares adjustment using the data from Problem 16.16 and the following additional data.
Control Stations
Unknown Stations
Station
X (m)
Y (m)
Station
X (m)
Y (m)
100
51,308.376
55,177.491
1
52,578.672
54,974.244
101
56,931.550
51,383.542
2
54,255.461
54,872.620
102
60,302.066
57,836.642
3
56,643.616
54,804.871
4
58,083.284
55,804.170
5
59,319.705
56,972.841
6
56,812.989
53,128.081
Distance Observations
Angle Observations
Course
Length (m)
S (m)
Angle
Value
S (″)
100–1
1286.464
±0.011
100–1–2
174°22′42″
±2.2
1–2
1679.859
±0.011
1–2–3
178°09′25″
±2.1
2–3
2389.108
±0.012
2–3–4
143°36′34″
±2.1
3–4
1752.497
±0.011
4–3–6
118°59′52″
±2.1
4–5
1701.349
±0.011
6–3–2
97°23′34″
±2.1
5–102
1308.130
±0.011
3–4–5
171°22′43″
±2.1
3–6
1685.342
±0.011
4–5–102
182°03′40″
±2.2
6–101
1748.563
±0.011
3–6–101
181°52′48″
±2.1
(a) What is the reference standard deviation, S0?
(b) List the adjusted coordinates of the unknown stations and their standard deviations.
(c) Tabulate the adjusted observations, their residuals, and their standard deviations.
(d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
Angle Observations
Angle
Value
S (″)
2–1–100
185°37′23″
±2.2
3–2–1
181°50′36″
±2.1
5–4–3
188°37′13″
±2.1
102–5–4
177°56′20″
±2.2
101–6–3
178°07′10″
±2.1
16.18 Perform an F test on the reference variances from Problem 16.16 and 16.17 at a 0.05 level of significance.
PROGRAMMING PROBLEMS
16.19 Write a computational program that reads a file of station coordinates and observations and then
(a) Writes the data to a file in a formatted fashion.
(b) Computes the J, K, and W matrices.
(c) Writes the matrices to a file that is compatible with the MATRIX program.
(d) Demonstrate this program with Problem 16.17.
16.20 Write a program that reads a file containing the J, K, and W matrices and then
(a) Writes these matrices in a formatted fashion.
(b) Performs one iteration in Problem 16.17.
(c) Writes the matrices used to compute the solution, and tabulates the corrections to the station coordinates in a formatted fashion.
16.21 Write a program that reads a file of station coordinates and observations and then
(a) Writes the data to a file in a formatted fashion.
(b) Computes the J, K, and W matrices.
(c) Performs a weighted least squares adjustment of Problem 16.17.
(d) Writes the matrices used in computations in a formatted fashion to a file.
(e) Computes the final adjusted station coordinates, their estimated errors, the adjusted observations, their residuals, and their estimated errors, and writes them to a file in a formatted fashion.
16.22 Develop a computational program that creates the coefficient, weight, and constant matrices for a network. Write the matrices to a file in a format usable by the MATRIX program supplied with this book. Demonstrate its use with Problem 16.17.