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

image 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.

Distance observation equation:

Angle observation equation:

Azimuth observation equation:

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.

Geometry for (a) Polygon and (b) link traverses.

FIGURE 16.1 (a) Polygon and (b) link traverses.

16.4 NUMERICAL EXAMPLE

16.5 MINIMUM AMOUNT OF CONTROL

image 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, images can be checked statistically. This check is often referred to as a goodness-of-fit test since the computation of images 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, images, 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 images 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 images 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.

TABLE 16.4 Two-Tailed χ2 Test on images

H0: S2 = 1
Ha: S2 ≠ 1
Test statistic:
images

Rejection region:
images
images

PROBLEMS

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.

  1. 16.1 Using the following control, adjust the data given for the Figure P16.1. The control station coordinates in units of feet are
    images
    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
    1. (a) What is the standard deviation of unit weight, S0?
    2. *(b) List the adjusted coordinates of station B and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
    Geometry for an incomplete rectangle.

    FIGURE P16.1

  2. 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
    1. *(a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
    Illustration of a geometrical triangle with the corners marked A, B, and C. A line is seen from Point A marked X.

    FIGURE P16.2

  3. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
    Illustration of a geometrical square with a line passing across from point A to point C.

    FIGURE P16.3

    The observed azimuth of line AB is 0°06′20.5″ ± 1.4″.

  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
    1. (a) What is the reference standard deviation, S0?
    2. *(b) List the adjusted coordinates of the unknown station and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  5. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown station and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  6. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown station and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  7. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  8. 16.8 Repeat Problem 16.7 using the additional data in Problem 15.10.
  9. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  10. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown station and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  11. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  12. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) List the inverted normal matrix used in the last iteration.
    5. (e) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  13. 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″
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  14. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  15. 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. 16.16 Using ADJUST, do a weighted least squares adjustment using the following data.
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (d) Perform a χ2 test on the reference variance at a 0.05 level of significance.
  17. 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
    1. (a) What is the reference standard deviation, S0?
    2. (b) List the adjusted coordinates of the unknown stations and their standard deviations.
    3. (c) Tabulate the adjusted observations, their residuals, and their standard deviations.
    4. (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
  18. 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

  1. 16.19 Write a computational program that reads a file of station coordinates and observations and then
    1. (a) Writes the data to a file in a formatted fashion.
    2. (b) Computes the J, K, and W matrices.
    3. (c) Writes the matrices to a file that is compatible with the MATRIX program.
    4. (d) Demonstrate this program with Problem 16.17.
  2. 16.20 Write a program that reads a file containing the J, K, and W matrices and then
    1. (a) Writes these matrices in a formatted fashion.
    2. (b) Performs one iteration in Problem 16.17.
    3. (c) Writes the matrices used to compute the solution, and tabulates the corrections to the station coordinates in a formatted fashion.
  3. 16.21 Write a program that reads a file of station coordinates and observations and then
    1. (a) Writes the data to a file in a formatted fashion.
    2. (b) Computes the J, K, and W matrices.
    3. (c) Performs a weighted least squares adjustment of Problem 16.17.
    4. (d) Writes the matrices used in computations in a formatted fashion to a file.
    5. (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.
  4. 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.

NOTE

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