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255

CHAPTER 15

ADJUSTMENT OF HORIZONTALSURVEYS: TRIANGULATION

15.1 INTRODUCTION

Prior to the development of electronic distance measuring equipment and theglobal positioning system, triangulation was the preferred method for extend-ing horizontal control over long distances. The positions of widely spacedstations were computed from measured angles and a minimal number of mea-sured distances called baselines. This method was used extensively by theNational Geodetic Survey in extending much of the national network. Tri-angulation is still used by many surveyors in establishing horizontal control,although surveys that combine trilateration (distance observations) with tri-angulation (angle observations) are more common. In this chapter, methods

are described for adjusting triangulation networks using least squares.A least squares triangulation adjustment can use condition equations or

observation equations written in terms of either azimuths or angles. In thischapter the observation equation method is presented. The procedure involvesa parametric adjustment where the parameters are coordinates in a plane rec-tangular system such as state plane coordinates. In the examples, the specifictypes of triangulations known as intersections, resections, and quadrilateralsare adjusted.

15.2 AZIMUTH OBSERVATION EQUATION

The azimuth equation in parametric form is

azimuth C (15.1)

Adjustment Computations: Spatial Data Analysis, Fourth Edition. C. D. Ghilani and P. R. Wolf 2006 John Wiley & Sons, Inc. ISBN: 978-0-471-69728-2

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256 ADJUSTMENT OF HORIZONTAL SURVEYS: TRIANGULATION

Figure 15.1 Relationship between the azimuth and the computed angle, .

TABLE 15.1 Relationship between the Quadrant, C, and the Azimuth

of the LineQuadrant Sign(xj xi) Sign(yj yi) Sign C Azimuth

I 0

II 180 180

III 180 180

IV 360 360

where tan1[(xj xi) / (yj yi)]; xi and yi are the coordinates of the

occupied station I; xj and yj are the coordinates of the sighted station J; andC is a constant that depends on the quadrant in which point J lies, as shownin Figure 15.1.

From the figure, Table 15.1 can be constructed, which relates the algebraicsign of the computed angle in Equation (15.1) to the value of C and thevalue of the azimuth.

15.2.1 Linearization of the Azimuth Observation Equation

Referring to Equation (15.1), the complete observation equation for an ob-served azimuth of line IJ is

x xj i1tan C Az v (15.2)ij Azijy yj i

where Azij is the observed azimuth, the residual in the observed azimuth,vAzijxi and yi the most probable values for the coordinates of station I, xj and yjthe most probable values for the coordinates of station J, and C a constant

with a value based on Table 15.1. Equation (15.2) is a nonlinear functioninvolving variables xi, yi, xj, and yj, that can be rewritten as

F(x,y ,x,y ) Az v (15.3)i i j j ij Azij

where

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15.2 AZIMUTH OBSERVATION EQUATION 257

x xj i1F(x,y ,x,y ) tan Ci i j j y yj i

As discussed in Section 11.10, nonlinear equations such as (15.3) can belinearized and solved using a first-order Taylor series approximation. Thelinearized form of Equation (15.3) is

F F FF(x,y ,x,y ) F(x,y ,x,y ) dx dy dx i i j j i i j j 0 i i jx y xi i j0 0 0

F dy (15.4)

j

yj 0

where (F/xi)0, (F/yi)0, (F/xj)0, and (F/yj)0 are the partial derivativesof F with respect to xi, yi, xj, and yj that are evaluated at the initial approxi-mations and and dxi, dyi, dxj, and dyj are the corrections appliedx , y , x , y ,i i j j0 0 0 0to the initial approximations after each iteration such that

x x dx y y dy x x dx y y dy (15.5)i i i i i i j j ji j j i0 0 0 0

To determine the partial derivatives of Equation (15.4) requires the prototypeequation for the derivative of tan1u with respect to x, which is

d 1 du1tan u (15.6)

2dx 1 u dx

Using Equation (15.6), the procedure for determining the F/xi is dem-onstrated as follows:

F 1 1

2x 1 [(x x) / (y y )] y yi j i j i j i

1(y y )j i (15.7)

2 2(x x) (y y )j i j i

y yi j

2IJ

By employing the same procedure, the remaining partial derivatives are

x x x y x xF F Fj i j i i j (15.8)

2 2 2y IJ x IJ y IJi j j

where IJ2 (xj xi)2

(yj yi)2.

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Figure 15.2 Relationship between an angle and two azimuths.

If Equations (15.7) and (15.8) are substituted into Equation (15.4) and theresults then substituted into Equation (15.3), the following prototype azimuthequation is obtained:

y y x x y y x xi j j i j i i jdx dy dx dy i i j j2 2 2 2IJ IJ IJ IJ

0 0 0 0

k v (15.9)Az Azij ij

Both

x xj i

1 2 2 2k Az tan C and IJ (x x) (y y )

Az ij j i 0 j i 0ij y yj i 0

are evaluated using the approximate coordinate values of the unknown pa-rameters.

15.3 ANGLE OBSERVATION EQUATION

Figure 15.2 illustrates the geometry for an angle observation. In the figure, Bis the backsight station, F the foresight station, and I the instrument station.As shown in the figure, an angle observation equation can be written as thedifference between two azimuth observations, and thus for clockwise angles,

x x x x i b i1 1BIF Az Az tan tan D vIF IB bif bify y y y i b i

(15.10)

where bif is the observed clockwise angle, the residual in the observedvbifangle, xb and yb the most probable values for the coordinates of the back-sighted station B, xi and yi the most probable values for the coordinates of

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15.3 ANGLE OBSERVATION EQUATION 259

the instrument station I, x and y the most probable values for the coordinatesof the foresighted station F, and D a constant that depends on the quadrantsin which the backsight and foresight occur. This term can be computed as

the difference between the C terms from Equation (15.1) as applied to thebacksight and foresight azimuths; that is,

D C Cif ib

Equation (15.10) is a nonlinear function of xb, yb, xi, yi, x, and y that canbe rewritten as

F(x ,y ,x,y ,x ,y ) v (15.11)b b i i bif

bif

where

x x x x i b i1 1F(x ,y ,x,y ,x ,y ) tan tan Db b i i y y y y i b i

Equation (15.11) expressed as a linearized first-order Taylor series expan-sion is

F FF(x ,y ,x,y ,x ,y ) F(x ,y ,x,y ,x ,y ) dx dy b b i i b b i i 0 b bx yb b0 0

F F F F dx dy dx dy i i x y x yi i 0 0 0 0

(15.12)

where F/xb, F/y

b, F/x

i, F/y

i, F/x

, and F/y

are the partial

derivatives of F with respect to xb, yb, xi, yi, x, and y, respectively.Evaluating partial derivatives of the function F and substituting into Equa-

tion (15.12), then substituting into Equation (15.11), results in the followingequation:

y yy y x x y y ii b b i b idx dy dx b b i2 2 2 2IB IB IB IF0 0 0

x x y y x xx xi i i i b dy dx dy (15.13)

i 2 2 2 2IB IF IF IF

0 0 0

k v bif bif

where

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Figure 15.3 Intersection example.

x x x x i b i1 1k tan tan D bif bif bif bif 0 0 y y y y i b i0 02 2 2 2 2 2IB (x x) (y y ) IF (x x) (y y )b i b i i i

are evaluated at the approximate values for the unknowns.In formulating the angle observation equation, remember that I is always

assigned to the instrument station, B to the backsight station, and F to theforesight station. This station designation must be followed strictly in em-ploying prototype equation (15.13), as demonstrated in the numerical exam-ples that follow.

15.4 ADJUSTMENT OF INTERSECTIONS

When an unknown station is visible from two or more existing control sta-tions, the angle intersection method is one of the simplest and sometimesmost practical ways for determining the horizontal position of a station. Fora unique computation, the method requires observation of at least two hori-zontal angles from two control points. For example, angles

1, and

2observed

from control stations R and Sin Figure 15.3, will enable a unique computationfor the position of station U. If additional control is available, computationsfor the unknown stations position can be strengthened by observing redun-dant angles such as angles

3and

4in Figure 15.3 and applying the method

of least squares. In that case, for each of the four independent angles, alinearized observation equation can be written in terms of the two unknowncoordinates, xu and yu.

Example 15.1 Using the method of least squares, compute the most prob-able coordinates of station U in Figure 15.3 by the least squares intersection

procedure. The following unweighted horizontal angles were observed fromcontrol stations R, S, and T:

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15.4 ADJUSTMENT OF INTERSECTIONS 261

500650 1013047 984117 5917011 2 3 4

The coordinates for the control stations R, S, and T are

x 865.40 x 2432.55 x 2865.22r s t

y 4527.15 y 2047.25 y 27.15r s t

SOLUTION

Step 1: Determine initial approximations for the coordinates of station U.

(a) Using the coordinates of stations R and S, the distance RS is computedas

2 2RS (2432.55 865.40) (4527.15 2047.25) 2933.58 ft

(b) From the coordinates of stations R and S, the azimuth of the linebetween R and S can be determined using Equation (15.2). Then theinitial azimuth of line RU can be computed by subtracting

1from the

azimuth of line RS:

x x 865.40 2432.55s r1 1Az tan C tan 180RS y y 4527.15 2047.25s r

147 4234

Az 1474234 500650 973544RU0

(c) Using the sine law with triangle RUS, an initial length for RU0

can becalculated as

RS sin 2933.58 sin(1003047 )2RU 6049.00 ft

0 sin(180 ) sin(282723 )1 2

(d) Using the azimuth and distance for RU0

computed in steps 1(b) and1(c), initial coordinates for station U are computed as

x x RU sin Az 865.40 6049.00 sin(973544 )u r 0 RU0 0

6861.35

y y RU cos Az 865.40 6049.00 cos(973544 )u r 0 RU0 0

3727.59

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(e) Using the appropriate coordinates, the initial distances for SU and TUare calculated as

2 2SU (6861.35 2432.55) (3727.59 2047.25)0

4736.83 ft

2 2TU (6861.35 2865.22) (3727.59 27.15)0

5446.29 ft

Step 2: Formulate the linearized equations. As in the trilateration adjustment,control station coordinates are held fixed during the adjustment by assign-ing zeros to their dx and dy values. Thus, these terms drop out of prototypeequation (15.13). In forming the observation equations, b, i, and areassigned to the backsight, instrument, and foresight stations, respectively,for each angle. For example, with angle

1, B, I, and F are replaced by U,

R, and S, respectively. By combining the station substitutions shown inTable 15.2 with prototype equation (15.13), the following observationequations are written for the four observed angles.

y y x xr u u r dx dy u u2 2RU RU0 0

x x x xs r u r 1 1 tan tan 0 v 1 1y y y ys r u r 0

y y x xu s s udx dy u u2 2SU SU 0 0

x x x xu s r s1 1 tan tan 0 v 2 2y y y yu s r s0 (15.14)

y y x xs u u sdx dy u u2 2SU SU 0 0

x x x xt s u s1 1 tan tan 180 v 3 3y y y yt s u s 0

y y x xu t t udx dy u u2 2TU TU 0 0x x x xu t s t 1 1

tan tan 0 v 4 4y y y yu t s t 0

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15.4 ADJUSTMENT OF INTERSECTIONS 263

TABLE 15.2 Substitutions

Angle B I F

1 U R S

2

R S U

3 U S T

4

S T U

Substituting the appropriate values into Equations (15.14) and multiplyingthe left side of the equations by to achieve unit consistency,1 the follow-ing J and K matrices are formed:

4527.15 3727.59 6861.35 865.402 26049.00 6049.00

3727.59 2047.25 2432.55 6861.35 4.507 33.8002 24736.83 4736.83 15.447 40.713

J 15.447 40.7132047.25 3727.59 6861.35 2432.55

2 2 25.732 27.7884736.83 4736.83

3727.59 27.15 2865.22 6861.352 25446.29 5446.29

2432.55 865.40 6861.35 865.401 1500650 tan tan 0

2047.25 4527.15 3727.59 4527.15

6861.35 2432.55 865.40 2432.551 11013047 tan tan 0

3727.59 2047.25 4527.15 2047.25K

2865.22 2432.55 6861.35 2432.551 1

984117

tan

tan

180

27.15 2047.25 3727.59 2047.25

6861.35 2865.22 2432.55 2865.221 1591701 tan tan 0

3727.59 27.15 2047.25 27.15

0.00

0.00

0.69 20.23

1 For these observations to be dimensionally consistent, the elements of the K and Vmatrices must

be in radian measure, or in other words, the coefficients of the K and J elements must be in the

same units. Since it is most common to work in the sexagesimal system, and since the magnitudes

of the angle residuals are generally in the range of seconds, the units of the equations are converted

to seconds by (1) multiplying the coefficients in the equation by , which is the number of seconds

per radian, or 206,264.8 /rad, and (2) computing the K elements in units of seconds.

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Notice that the initial coordinates for and were calculated using 1

x yu u0 0and

2, and thus their K-matrix values are zero for the first iteration. These

values will change in subsequent iterations.

Step 3: Matrix solution. The least squares solution is found by applyingEquation (11.37).

1159.7 1820.5TJ J

1820.5 5229.7

0.001901 0.000662T 1Q (J J) xx 0.000662 0.000422

509.9TJ K 534.1

0.001901 0.000662 509.9 dxT 1 T uX (J J) (J K) 0.000662 0.000422 534.1 dyu

dx 0.62 ft and dy 0.11 ftu u

Step 4: Add the corrections to the initial coordinates for station U:

x x dx 6861.35 0.62 6860.73u u u0 (15.15)

y y dy 3727.59 0.11 3727.48u u u0

Step 5: Repeat steps 2 through 4 until negligible corrections occur. The nextiteration produced negligible corrections, and thus Equations (15.15) pro-duced the final adjusted coordinates for station U.

Step 6: Compute post-adjustment statistics. The residuals for the angles are

4.507 33.80 0.0015.447 40.713 0.62 0.00

V JX K 15.447 40.713 0.11 0.69

25.732 27.788 20.23

6.55.1

5.8

7.3

The reference variance (standard deviation of unit weight) for the adjust-ment is computed using Equation (12.14) as

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15.5 ADJUSTMENT OF RESECTIONS 265

6.55.1

TV V [6.5 5.1 5.8 7.3] [155.2]5.8

7.3

TV V 155.2S 8.8

0 m n 4 2The estimated errors for the adjusted coordinates of station U, given byEquation (13.24), are

S S Q 8.8 0.001901 0.38 ftx 0 x xu u u

S S Q 8.8 0.000422 0.18 fty 0 y yu u u

The estimated error in the position of station U is given by

2 2 2 2S S S 0.38 0.18 0.42 ftu x y

15.5 ADJUSTMENT OF RESECTIONS

Resection is a method used for determining the unknown horizontal positionof an occupied station by observing a minimum of two horizontal angles toa minimum of three stations whose horizontal coordinates are known. If morethan three stations are available, redundant observations are obtained and theposition of the unknown occupied station can be computed using the least

squares method. Like intersection, resection is suitable for locating an occa-sional station and is especially well adapted over inaccessible terrain. Thismethod is commonly used for orienting total station instruments in locationsfavorable for staking projects by radiation using coordinates.

Consider the resection position computation for the occupied station U ofFigure 15.4 having observed the three horizontal angles shown between sta-tions P, Q, R, and S whose positions are known. To determine the positionof station U, two angles could be observed. The third angle provides a checkand allows a least squares solution for computing the coordinates of sta-

tion U.Using prototype equation (15.13), a linearized observation equation can bewritten for each angle. In this problem, the vertex station is occupied and isthe only unknown station. Thus, all coefficients in the Jacobian matrix followthe form used for the coefficients of dxi and dyi in prototype equation (15.13).

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Figure 15.4 Resection example.

The method of least squares yields corrections, dxu and dyu, which gives themost probable coordinate values for station U.

15.5.1 Computing Initial Approximations in the Resection Problem

In Figure 15.4 only two angles are necessary to determine the coordinates ofstation U. Using stations P, Q, R, and U, a procedure to find the station Usapproximate coordinate values is

Step 1: Let

QPU URQ G 360 (1 2 RQP) (15.16)

Step 2: Using the sine law with triangle PQU yields

QU PQ (a)

sin QPU sin 1

and with triangle URQ,

QU QR (b)

sin URQ sin 2

Step 3: Solving Equations (a) and (b) for QU and setting the resulting equa-tions equal to each other gives

PQ sin PQU QR sin URQ (c)sin 1 sin 2

Step 4: From Equation (c), let H be defined as

sin QPU QR sin 1H (15.17)

sin URQ PQ sin 2

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Step 5: From Equation (15.16),

QPU G URQ (d)

Step 6: Solving Equation (15.17) for the sinQPU, and substituting Equation(d) into the result gives

sin(G URQ) H sinURQ (e)

Step 7: From trigonometry

sin( ) sin cos cos sin

Applying this relationship to Equation (e) yields

sin G URQ sin G cosURQ cos G sinURQ ()

sin G URQ H sin URQ (g)

Step 8: Dividing Equation (g) by cos URQ and rearranging yields

sin G tanURQ[H cos(G)] (h)

Step 9: Solving Equation (h) for URQ gives

sin G1

URQ tan (15.18)H cos G

Step 10: From Figure 15.4,

RQU 180 (2 URQ) (15.19)

Step 11: Again applying the sine law yields

QR sin RQURU (15.20)

sin 2

Step 12: Finally, the initial coordinates for station U are

x x RU sin(Az URQ)u r RQ (15.21)

y y RU cos(Az URQ)u r RQ

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Example 15.2 The following data are obtained for Figure 15.4. Controlstations P, Q, R, and S have the following (x,y) coordinates: P (1303.599,1458.615), Q (1636.436, 1310.468), R (1503.395, 888.362), and S (1506.262,

785.061). The observed values for angles 1, 2, and 3 with standard deviationsare as follows:

Backsight Occupied Foresight Angle S ( )

P U Q 302933 5

Q U R 383031 6

R U S 102957 6

What are the most probable coordinates of station U?

SOLUTION Using the procedures described in Section 15.5.1, the initialapproximations for the coordinates of station U are:

(a) From Equation (15.10),

RQP Az Az 2935938.4 1972938.4PQ QR

963000.0

(b) Substituting the appropriate angular values into Equation (15.16) gives

G 360 (302933 383031 963000.0 ) 1942956

(c) Substituting the appropriate station coordinates into Equation (14.1)yields

PQ 364.318 and QR 442.576

(d) Substituting the appropriate values into Equation (15.17) yields H as

442.576 sin(302933 )H 0.990027302

364.318 sin(383031 )

(e) Substituting previously determined G and H into Equation (15.18),URQ is computed as

sin(1942956 )1

URQ tan 1800.990027302 cos(1942956 )

850022 180 945936.3

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(f) Substituting the value ofURQ into Equation (15.19), RQU is de-termined to be

RQU 180 (383031 945936.3 ) 462952.7

(g) From Equation (15.20), RU is

442.576 sin(462952.7 )RU 515.589

sin(383031 )

(h) Using Equation (15.1), the azimuth of RQ is

1636.436 1503.3951Az tan 0 172938.4RQ 1310.468 888.362

(i) From Figure 15.4, AzRU is computed as

Az 1972938.4 180 172938.4RQ

Az Az URQ 360 172938.4 945936.3RU RQ

2823002.2

( j) Using Equation (15.21), the coordinates for station U are

x 1503.395 515.589 sin Az 1000.03u RU

y 888.362 515.589 cos Az 999.96u RU

For this problem, using prototype equation (15.13), the J and K matricesare

y y y y x x x xp u q u u p u q 2 2 2 2UP UQ UP UQ

0 0

y y x xy y x xq u u qr u u r J 2 2 2UQ UR UQ UR0 0

y y y y x x x xr u s u u r u s

2 2 2 2UR US UR US 0 0(1 1 )

0

K (2 2 )0 (3 3 )0

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Also, the weight matrix W is a diagonal matrix composed of the inverses ofthe variances of the angles observed, or

10 0

251

W 0 026 1

0 026

Using the data given for the problem together with the initial approxima-tions computed, numerical values for the matrices were calculated and theadjustment performed using the program ADJUST. The following results wereobtained after two iterations. The reader is encouraged to adjust these exampleproblems using both the MATRIX and ADJUST programs supplied.

ITERATION 1

J MATRIX K MATRIX X MATRIX

====================== ======== ========

184.993596 54.807717 0.203359 0.031107

214.320813 128.785353 0.159052 0.065296

59.963802 45.336838 6.792817

ITERATION 2

J MATRIX K MATRIX X MATRIX

====================== ======== ========

185.018081 54.771738 1.974063 0.000008

214.329904 128.728773 1.899346 0.000004

59.943758 45.340316 1.967421

INVERSE MATRIX

=======================

0.00116318 0.00200050

0.00200050 0.00500943

Adjusted stations

Station X Y Sx Sy

===========================================

U 999.999 1,000.025 0.0206 0.0427

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15.6 ADJUSTMENT OF TRIANGULATED QUADRILATERALS 271

Adjusted Angle Observations

Station Station Station

Backsighted Occupied Foresighted Angle V S ( )========================================================

P U Q 30 29 31 2.0 2.3

Q U R 38 30 33 1.9 3.1

R U S 10 29 59 2.0 3.0

Redundancies 1

Reference Variance 0.3636

Reference So 0.60

15.6 ADJUSTMENT OF TRIANGULATED QUADRILATERALS

The quadrilateral is the basic figure for triangulation. Procedures like thoseused for adjusting intersections and resections are also used to adjust this

figure. In fact, the parametric adjustment using the observation equationmethod can be applied to any triangulated geometric figure, regardless of itsshape.

The procedure for adjusting a quadrilateral consists of first using a mini-mum number of the observed angles to solve the triangles, and computinginitial values for the unknown coordinates. Corrections to these initial coor-dinates are then calculated by applying the method of least squares. Theprocedure is iterated until the solution converges. This yields the most prob-able coordinate values. A statistical analysis of the results is then made. The

following example illustrates the procedure.

Example 15.3 The following observations are supplied for Figure 15.5. Ad-just this figure by the method of unweighted least squares. The observedangles are as follows:

1 423529.0 3 795442.1 5 212923.9 7 312045.8

2 873510.6 4 182822.4 6 390135.4 8 393427.9

The fixed coordinates are

x 9270.33 y 8448.90 x 15,610.58 y 8568.75A A D D

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Figure 15.5 Quadrilateral.

TABLE 15.3 Structure of the Coefficient or J Matrix in Example 15.3

Angle

Unknowns

dxb dyb dxc dyc

1 dx(b) dy(b) dx() dy()

2 0 0 dx(b) dy(b)

3 dx(i) dy(i) dx(b) dy(b)

4 dx(i) dy(i) 0 0

5 0 0 dx(i) dy(i)

6 dx() dy() dx(i) dy(i)

7 dx() dy() 0 0

8 dy(b) dy(b) dx() dy()

SOLUTION The coordinates of stations B and C are to be computed in thisadjustment. The Jacobian matrix has the form shown in Table 15.3. The sub-scripts b, i, and of the dxs and dys in the table indicate whether stations

B and Care the backsight, instrument, or foresight station in Equation (15.13),respectively. In developing the coefficient matrix, of course, the appropriatestation coordinate substitutions must be made to obtain each coefficient.

A computer program has been used to form the matrices and solve theproblem. In the program, the angles were entered in the order of 1 through8. The X matrix has the form

dxbdybX dxc dyc

The following self-explanatory computer listing gives the solution for thisexample. As shown, one iteration was satisfactory to achieve convergence,since the second iteration produced negligible corrections. Residuals, adjusted

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15.6 ADJUSTMENT OF TRIANGULATED QUADRILATERALS 273

coordinates, their estimated errors, and adjusted angles are tabulated at theend of the listing.

*******************************************

Initial approximations for unknown stations

*******************************************

Station X Y

==============================

B 2,403.600 16,275.400

C 9,649.800 24,803.500

Control Stations

Station X Y

==============================

A 9,270.330 8,448.900

D 15,610.580 8,568.750

******************

Angle Observations

******************


Backsighted Occupied Foresighted Angle

===============================================

B A C 42 35 29.0

C A D 87 35 10.6

C B D 79 54 42.1

D B A 18 28 22.4

D C A 21 29 23.9

A C B 39 01 35.4

A D B 31 20 45.8

B D C 39 34 27.9

Iteration 1

J Matrix K MATRIX X MATRIX

---------------------------------------------- --------- -----------

14.891521 13.065362 12.605250 0.292475 1.811949 1 0.011149

0.000000 0.000000 12.605250 0.292475 5.801621 2 0.049461

20.844399 0.283839 14.045867 11.934565 3.508571 3 0.061882

8.092990 1.414636 0.000000 0.000000 1.396963 4 0.036935

0.000000 0.000000 1.409396 4.403165 1.833544

14.045867 11.934565 1.440617 11.642090 5.806415

6.798531 11.650726 0.000000 0.000000 5.983393

6.798531 11.650726 11.195854 4.110690 1.818557

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Iteration 2

J Matrix K MATRIX X MATRIX

---------------------------------------------- --------- -----------14.891488 13.065272 12.605219 0.292521 2.100998 1 0.000000

0.000000 0.000000 12.605219 0.292521 5.032381 2 0.000000

20.844296 0.283922 14.045752 11.934605 4.183396 3 0.000000

8.092944 1.414588 0.000000 0.000000 1.417225 4 0.000001

0.000000 0.000000 1.409357 4.403162 1.758129

14.045752 11.934605 1.440533 11.642083 5.400377

6.798544 11.650683 0.000000 0.000000 6.483846

6.798544 11.650683 11.195862 4.110641 1.474357

INVERSE MATRIX

-------------------------------

0.00700 0.00497 0.00160 0.01082

0.00497 0.00762 0.00148 0.01138

0.00160 0.00148 0.00378 0.00073

0.01082 0.01138 0.00073 0.02365

*****************

Adjusted stations

*****************

Station X Y Sx Sy

================================================

B 2,403.589 16,275.449 0.4690 0.4895

C 9,649.862 24,803.537 0.3447 0.8622

***************************

Adjusted Angle Observations

***************************


Backsighted Occupied Foresighted Angle V S

===============================================================

B A C 42 35 31.1 2.10 3.65

C A D 87 35 15.6 5.03 4.33

C B D 79 54 37.9 4.18 4.29

D B A 18 28 21.0 1.42 3.36

D C A 21 29 25.7 1.76 3.79

A C B 39 01 30.0

5.40 4.37A D B 31 20 52.3 6.48 4.24

B D C 39 34 26.4 1.47 3.54

*********************************

Adjustment Statistics

********************************

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PROBLEMS 275

Iterations 2

Redundancies 4

Reference Variance 31.42936404

Reference So 5.6062

Convergence!

PROBLEMS

15.1 Given the following observations and control station coordinates to

accompany Figure P15.1, what are the most probable coordinates forstation E using an unweighted least squares adjustment?

Figure P15.1

Control stations

Station X (ft) Y (ft)

A 10,000.00 10,000.00

B 11,498.58 10,065.32

C 12,432.17 11,346.19D 11,490.57 12,468.51

Angle observations

Backsight, b Occupied, i Foresight, Angle S ( )

E A B 905957 5.3

A B E 402602 4.7

E B C 880855 4.9

B C E 524502 4.7E C D 510955 4.8

C D E 931314 5.0

15.2 Repeat Problem 15.1 using a weighted least squares adjustment withweights of 1/S2 for each angle. What are:

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(a) the most probable coordinates for station E?

(b) the reference standard deviation of unit weight?

(c) the standard deviations in the adjusted coordinates for station E?(d) the adjusted angles, their residuals, and the standard deviations?

15.3 Given the following observed angles and control coordinates for theresection problem of Figure 15.4:

1 494703 2 332155 3 475853

Assuming equally weighted angles, what are the most probable co-ordinates for station U?

Control stations

Station X (m) Y (m)

P 2423.077 3890.344

Q 3627.660 3602.291

R 3941.898 2728.314

S 3099.018 1858.429

15.4 If the estimated standard deviations for the angles in Problem 15.3are S

1 3.1 , S

2 3.0 , and S

3 3.1 , what are:

(a) the most probable coordinates for station U?


(c) the standard deviations in the adjusted coordinates of station U?

(d) the adjusted angles, their residuals, and the standard deviations?

15.5 Given the following control coordinates and observed angles for anintersection problem:

Control stations

Station X (m) Y (m)

A 100,643.154 38,213.066

B 101,093.916 67,422.484

C 137,515.536 67,061.874

D 139,408.739 37,491.846

Angle observationsBacksight Occupied Foresight Angle S ( )

D A E 3193950 5.0

A B E 3052117 5.0

B C E 3225035 5.0

C D E 3131022 5.0

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PROBLEMS 277

What are:

(a) the most probable coordinates for station E?

(b) the reference standard deviation of unit weight?(c) the standard deviations in the adjusted coordinates of station E?


15.6 The following control station coordinates, observed angles, and stan-dard deviations apply to the quadrilateral in Figure 15.5.

Control stations Initial approximations

Station X (ft) Y (ft) Station X (ft) Y (ft)

A 2546.64 1940.26 B 2243.86 3969.72

D 4707.04 1952.54 C 4351.06 4010.64

Angle observations


B A C 493330 4.2

C A D 483554 4.2

C B D 402544 4.2D B A 421156 4.2

D C A 505307 4.2

A C B 474847 4.2

A D B 393834 4.2

B D C 405220 4.2

Do a weighted adjustment using the standard deviations to calculate

weights. What are:(a) the most probable coordinates for stations B and C?


(c) the standard deviations in the adjusted coordinates for stations Band C?


Figure P15.7

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15.7 For Figure P15.7 and the following observations, perform a weightedleast squares adjustment.

(a) Station coordinate values and standard deviations.

(b) Angles, their residuals, and the standard deviations.


Station X (m) Y (m) Station X (m) Y (m)

A 114,241.071 91,294.643 C 135,982.143 107,857.143

B 116,607.143 108,392.857 D 131,567.500 90,669.643

Angle observations


B A C 444915.4 2.0

C A D 392158.0 2.0

C B D 481448.9 2.0

D B A 480249.6 2.0

D C A 381738.0 2.0

A C B 385303.9 2.0

A D B 474556.8 2.0

B D C 543426.1 2.0

15.8 Do Problem 15.7 using an unweighted least squares adjustment. Com-pare and discuss any differences or similarities between these resultsand those obtained in Problem 15.7.

15.9 The following observations were obtained for the triangulation chainshown in Figure P15.9.


Station X (m) Y (m) Station X (m) Y (m)

A 103,482.143 86,919.643 C 103,616.071 96,116.071

B 118,303.571 86,919.643 D 117,991.071 95,580.357

G 104,196.429 112,589.286 E 104,375.000 104,196.429

H 118,080.357 112,767.857 F 118,169.643 104,598.214

Angle observations

B I F Angle S ( ) B I F Angle S ( )

C A D 581952 3 D A B 304956 3

A B C 320311 3 C B D 555251 3

D C B 295501 3 B C A 584653 3

B D A 611402 3 A D C 325806 3

E C F 542400 3 F C D 322205 3

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PROBLEMS 279

C D E 301127 3 E D F 584832 3

D E C 630221 3 F E D 335936 3

D F C 583750 3 C F E 283400 3

H E F 302108 3 G E H 591148 3E F G 312555 3 G F H 593631 3

H G F 303001 3 F G E 590104 3

F H E 583708 3 E H G 311711 3

Figure P15.9

Use ADJUST to perform a weighted least squares adjustment. Tab-ulate the final adjusted:

(a) station coordinates and their standard deviations.

(b) angles, their residuals, and the standard deviations.

15.10 Repeat Problem 15.9 using an unweighted least squares adjustment.Compare and discuss any differences or similarities between theseresults and those obtained in Problem 15.9. Use the program ADJUST

in computing the adjustment.

15.11 Using the control coordinates from Problem 14.3 and the followingangle observations, compute the least squares solution and tabulatethe final adjusted:


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Angle observations


B A C 2804106 5.2

C A D 392153 5.1

C B D 513616 5.2

D B A 2555003 5.2

D C A 1012717 5.2

A C B 3115238 5.2

A D B 3240704 5.1

B D C 750350 5.2

15.12 The following observations were obtained for a triangulation chain.


Station X (ft) Y (ft) Station X (ft) Y (ft)

A 92,890.04 28,566.74 B 93,611.26 47,408.62

D 93,971.87 80,314.29 C

E

93,881.71

111,191.00

64,955.36

38,032.76

F 110,109.17 57,145.10G 110,019.02 73,102.09

H 131,475.32 28,837.20I 130,213.18 46,777.56

J 129,311.66 64,717.91K 128,590.44 79,142.31

Angle observations

B I F Angle S ( ) B I F Angle S ( )

B A E 602728 2.2 E B A 640706 2.1F B E 583714 2.1 C B F 583409 2.1

F C B 651051 2.3 G C F 522914 2.0D C G 625242 2.1 G D C 660808 2.2

D G K 1374657 2.4 K G J 413018 2.7J G F 661115 2.1 F G C 633213 2.1

C G D 505911 2.3 B F C 561456 2.6C F G 635829 2.1 G F J 684805 2.1

J F I 484805 2.3 I F E 592849 2.4E F B 624129 2.2 A E B 552519 2.3

B E F 584113 2.5 F E I 683306 2.0I E H 490427 2.1 H E A 1281552 2.6

E H I 613524 2.5 H I E 692010 2.0

E I F 515806 2.1 F I J 595035 2.2

I J F 712112 2.2 F J G 450039 2.1

G J K 633857 2.2 J K G 745046 2.3

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PROBLEMS 281

Use ADJUST to perform a weighted least squares adjustment. Tab-ulate the final adjusted:



15.13 Do Problem 15.12 using an unweighted least squares adjustment.Compare and discuss any differences or similarities between theseresults and those obtained in Problem 15.12. Use the program AD-JUST in computing the adjustment.

Use the ADJUST software to do the following problems.

15.14 Problem 15.2

15.15 Problem 15.4

15.16 Problem 15.5

15.17 Problem 15.6

15.18 Problem 15.9

Programming Problems15.19 Write a computational program that computes the coefficients for pro-

totype equations (15.9) and (15.13) and their k values given the co-ordinates of the appropriate stations. Use this program to determinethe matrix values necessary to do Problem 15.6.

15.20 Prepare a computational program that reads a file of station coordi-nates, observed angles, and their standard deviations 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) test this program with Problem 15.6.

15.21 Write a computational 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 of either a weighted or unweighted least

squares adjustment of Problem 15.6.(c) writes the matrices used to compute the solution and the correc-

tions to the station coordinates in a formatted fashion.

15.22 Write a computational program that reads a file of station coordinates,observed angles, and their standard deviations and then:

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(a) writes the data to a file in a formatted fashion.

(b) computes the J, K, and W matrices.

(c) performs either a relative or equal weight least squares adjustmentof Problem 15.6.

(d) writes the matrices used to compute the solution and tabulatesthe final adjusted station coordinates and their estimated errorsand the adjusted angles, together with their residuals and theirestimated errors.

15.23 Prepare a computational program that solves the resection problem.Use this program to compute the initial approximations for Problem15.3.

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CHAPTER 16

ADJUSTMENT OF HORIZONTAL

SURVEYS: TRAVERSESAND NETWORKS

16.1 INTRODUCTION TO TRAVERSE ADJUSTMENTS

Of the many methods that exist for traverse adjustment, the characteristic thatdistinguishes 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 thetraverse but provide the most probable values for the given data set. Addi-tionally, the observations can be rigorously weighted based on their estimatederrors and adjusted accordingly. Given these facts, together with the compu-tational power now provided by computers, it is hard to justify not using leastsquares for all traverse adjustment work.

In this chapter we describe methods for making traverse adjustments byleast squares. As was the case in triangulation adjustments, traverses can beadjusted by least squares using either observation equations or conditionalequations. Again, because of the relative ease with which the equations canbe written and solved, the parametric observation equation approach is dis-cussed.

16.2 OBSERVATION EQUATIONS

When adjusting a traverse using parametric equations, an observation equationis written for each distance, direction, or angle. The necessary linearizedobservation equations developed previously are recalled in the followingequations.

adj_tota

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