the adjustment of some geodetic networks using microsoft excel

8/13/2019 The Adjustment of Some Geodetic Networks Using Microsoft EXCEL

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Turkish Journal of Science & TechnologyVolume 4, No 2, 139- 155, 2009

The Adjustment of Some Geodetic Networks Using Microsoft EXCELSOLVER

Levent TAI

Frat University, Engineering Faculty, Civil Engineering Department, 23100, Elaz, [email protected]

(Received: 20.06.2009; Accepted: 21.09.2009)

AbstractThe Solver is an Excel Add-in that is developed to solve mathematical equations and perform optimizationsolutions for modeled systems as well. In this study, Excel Solver Add-in that is developed by Frontline Systemsis used to solve the adjustment problems by using least squares method. As case studies, adjustment of the closed

and open traverses, leveling networks and GPS coordinates are selected

Key word: Adjustment, Solver, Geodetic Networks, Leveling, GPS

Microsoft EXCEL SOLVER Kullanlarak BazJeodezik Alarn Dengelenmesi

zetSolver modellenmisistemlerin optimizasyon problemlerinin zm ile matematiksel denklemleri zmek iingelitirilmi bir Excel eklentisidir. Bu almada, Frontline Sistemleri tarafndan gelitirilmi Excel Solvereklentisi en kk kareler yntemi kullanlarak dengeleme problemlerini zmek iin kullanlmtr. almaolarak ak ve kapalpoligonlar ile nivelman ave GPS koordinatlarnn dengelenmesi seilmitir.

Anahtar Szckler:Dengeleme, zc, Jeodezik Alar, Nivelman, GPS

1.Introduction

According to zturk [1], data adjustmentis to determine the adjusted or estimated valuesof the unknowns by using all redundantmeasurements, which are in excess of thenumber of measurements that are required touniquely determine the underlying model. Themain purpose of least squares method is todetermine the accuracy and reliability ofobservations, functions of observations andestimated values. The Solver, to adjust small tomedium size traverses encountered in day to daysurveying practices. Furthermore, the user doesnot need to know how to form the conditionequations how to differentiate complexequations how to solve the normal equationsused in a conventional least squares approach[2]..

In this study, usability of Excel SolverAdd-in, developed by Frontline systems, isinvestigated for analysis and adjustment of

survey measurements using least squarestechnique. Excel Solver Add-in and Visual basicscripts are used to estimate the adjustedcoordinates of closed and open traverses,levelling networks and GPS coordinates. Thedata adjustment examples used in this study wasfirst solved using conventional least squaresmethod. Then, same examples are re-solvedusing Excel solver and the results from the twomethods are compared and discussed.

2. About the EXCEL SOLVER

Microsoft Excel Solver is a MicrosoftExcel add-in. Microsoft Excel Solver helps youto determine the optimum value for a formula ina particular target cell on a Microsoft Excelworksheet. Microsoft Excel Solver adjusts thevalues of other cells that are related to the targetcell by using an equation. After you construct anequation and define a set of parameters orconstraints for the variables in the equation,

Microsoft Excel Solver tries various solutions to


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arrive at an answer that satisfies all constraints[3]. For additional information the reader isencouraged to visit the Frontline web site at

http://www.frontsys.com. The site contains linksto the most current information on the Solverplatforms, examples, tutorials on differentoptimization models, although not much on leastsquares application to surveying problems.

2.1 Installing Microsoft EXCELS SOLVERTo install Solver, select Tools/Add-Ins.

Excel displays a dialogue box with a list of allAdd-Ins currently available as shown in figure 1.Scroll down the list and check the box next toSolver add-in. Click OK. Excel will install the

add-in and place a new menu item (Solver)under the Tools menu.

Fig. 1. The Add-Ins dialog box [6].

If Solver add-in does not appear on the list, thento install Solver add-in must use MicrosoftOffice CD-ROM.

Fig. 2. Solver Parameters Dialog Box [7]

2.2 The Solver Parameters

On the Tools menu, click Solver. Exceldisplays a solver parameters dialogue box asshown in figure 2.

Microsoft Excel solver uses the followingelements to solve an equation.

The target cell is reference for the objectivefunction. It is the cell in the worksheet modelthat will be minimized, maximized, or set to acertain value.

Changing cell are decision variables. Theseadjustable cells must be related directly orindirectly to the target cell.

Constraints are restrictions on the contentsof cells. For each constraint do the following:

1. Click ADD button. This will bring up thedialog box in figure 3.

2. In the cell reference box, enter the right handside of the constraint.

3. In the cell reference box,enter the left handside of theset of Cell Reference.

4. In the middle box should be chosen one of(=, =)

Fig. 3.Add Constraint Dialog Box [7]

Click options button in the Solver ParametersDialog Box. This will bring up the SolverOptions dialog box in figure 4. In the standardExcel Solver, all such options appear in onedialog box; in the Premium Solver products,where many more options and tolerances areavailable, each optimizer has a separate dialog

box [4]. The Max Time and the Iterations editboxes control the Solvers running time. TheShow Iteration Results check box instructs theSolver to pause after each major iteration anddisplay the current trial solution on thespreadsheet. In lieu of these options, however,the user can simply press the ESC key at anytime to interrupt the Solver, inspect the currentiterate, and decide whether to continue or tostop. The Assume Linear Model check boxdetermines whether the simplex method or theGRG2 nonlinear programming algorithm will

be used to solve the problem. The Use

Objective Function Cell Location

Min,Max orValue

Decision

VariableCell Locations

SolveProble

Constraint Set

ClearCurrent

ModelAdd a ConstraintChange a ConstraintDelete a Constraint

InvokeSolverOptions

Constraint CellReference

or set of CellReference Constraint Type

Formula, Cell Reference,

Or Value


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The Adjustment of Some Geodetic Networks Using Microsoft Excel SOLVER

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Automatic Scaling check box causes the modelto be rescaled internally before solution. TheAssume Non-Negative check box places lower

bounds of zero on any decision variables that donot have explicit bounds in the Constraints listbox.

The Precision edit box is used by all of theoptimizers and indicates the tolerance withinwhich constraints are considered binding andvariables are considered integral in mixed-integer-programming (MIP) problems. TheTolerance edit box (a somewhat unfortunatename, but Microsofts choice) is the integeroptimality or MIP gap tolerance used in the

branch-and bound method. The GRG2 algorithm

uses the Convergence edit box and Estimates,Derivatives, and Search option button groups[4].

Fig. 4.Solver Options Dialog Box [7].

Click solve button. Excel displays a SolverResults dialogue box as shown in figure 5 amessage appears on the top left-hand side ofthe box. In this case, Excel reports that Solverfound a solution. All constraints and optimallyconditions are satisfied.

Fig. 5. Solver Results Box [7].

3. The Essential of Adjustment by LeastSquare Method

The adjustment is based on equations

where the observations are expressed as functionof unknown parameters. A Taylor seriesexpansion is usually performed in the case of anonlinear relationship between observations andunknowns. The resulting linear relationship can

be represented in a matrix - vector notation as

l= A * x (1)Where;l ... vector of observation, A ... design matrix, x... vector of unknownsn observations and u unknown parameters lead

to a design matrix A comprising n rows and ucolumns.We have to add a noise vector v to theobservations.

l + v = A * x (2)

The job of the adjustment is to find the set ofunknowns which result in the smallest noisevector v. A possible criterion for the noise vectorto be small is that its norm is small.

vT* v = minimum. (3)

This is the principle of least squares. Thesolution vector x of the unknown coordinates isthen computed as:

x = (AT* A)-1* AT* l (4)

In addition, the observations are weightedaccording to their variance - covariance. Theweighting factor of an observation is thereciprocal of its variance, which, in turn, is the

square of its standard deviation )Q(P -1ll= .Thatmeans that observations that is considered to bevery accurate (small standard deviation) gets a

big weighting factor and therefore stronglyaffects the adjustment result. On the other hand,giving an observation a big standard deviation(relatively to the other observations) makes itsimpact on the adjustment result small. Then, weminimize the quantity

vT* P * v = minimum. (5)

IP Solutionwithin this %of Optimal

UseSimplexAlgoritma

Assume Non-Negative

Nonlinearoptions


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(P is the weighting matrix of the observations).The solution vector x of the unknowncoordinates is then computed as:

x = (A

T

* P * A)

-1

* A

T

* P * l (6)

4. The Regulations to Various GeodeticNetworks

4.1Leveling NetworksThere are two different cases for levelingnetworks:

4.1.1 Leveling Network Connecting TwoFixed Points

Fig. 6. Leveling Network Connecting Two FixedPoints [5]

(7)

(8)(9)

Where;HA, HB, HCand HDare point heights. h1, h2andh3 are height differences between two neighbor

pointsNumber of redundant observations= r = n-u(degrees of freedom)n= number of observation (measurements), u=number of unknowns

4.1.2 Closed Leveling

Fig. 7. Closed Leveling [5].

(10)

(11)

(12)Where; HA, HB and HC are point heights. h1, h2and h3are height differences measured betweentwo points

Number of redundant observations= r = n-u(degrees of freedom)n= number of observation (measurements), u=number of unknowns

4.2 Traverse Networks4.2.1 Traverse Connecting Two Fixed Points

Fig. 8. Traverse Connecting Two Fixed Points [5].

(13)

(14)

(15)

(16)

(17)

(18)

Where; xi, yi are coordinates of points.i= azimuth ,di= distances

Number of redundant observations= r = n-u(degrees of freedom)n= number of observation (measurements), u=number of unknowns

HA

HB

HCHD

h3

h2h1

HA

HB

HC

h3

h2

h1

2

A

B3

2

1

Fixed Point

Fixed Point

d1 d2 d3

246

46

sin

cos

sin0)(2

1

cos0)(2

1

3

3

2

21

1

2

2

1

1

==

==

+=

+=

===

===

==

r

un

iidAyiy

iidAxix

iidiyayby

iiy

iidixaxbx

iix

d

d

d

l

y

x

y

x

x

0)(321

1

]321[3,1][2,1

=+++

=

==

BHAHhhh

r

hhhT

LDHCHT

X

0

1

][][

321

3213,12,1

=

=

==

hhh

r

hhhLHHXT

CB

T


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143

4.2.2 Closed Traverse

Fig. 9.Closed traverse [5].

(19)

(20)

(21)

(22)

Where;xiand yiare coordinates of points. i= azimuth ,

di= distancesNumber of redundant observations= r = n-u(degrees of freedom)n= number of observation (measurements),u= number of unknowns

4.3Adjustment Of Coordinates DeterminedFrom Satellite Observation

Today, the geocentric coordinates ofgeodetic networks can be determined usingsatellite observations (GPS Measurements).

Coordinates of the geodetic points can bedetermined using the measurements done in oneor more surveying sessions, and thesemeasurements are needs to be adjusted. Theadjusted coordinates are calculated bydetermining the adjusted components ofunknown vectors. The components of theunknown vectors are made up of the coordinatevalues of the points to be adjusted. Theaccuracies of the components may be differentand they may have physical and/or mathematicalcorrelation. Adjusted values of unknown vectors

that were calculated more than once in different

times are determined using Adjustment ofUnknown vectors that have different accuracyand correlation approach [1].

4.3.1 Adjustment of Unknown Vectors thathave Different Accuracy and Correlation

Functional model

Condition equations are set as;

(23).

.

Where;v = estimated residuals (estimated error), x =estimated value, Li= observed value

Stochastic Model

(24)

Where;Qi = Cofactor matrices, Ki = Variance-Covariance matrices,

errormeanofsquare20=m , Pi = Weights

Objective Function

(25)

Where;v = estimated residuals (estimated error),Pi = Weights

Normal equations

(26)

The Adjustment Unknowns

(27)

d

d

d

d

268

68

)1(

)1(tan)(

)22

1()22

1(

==

==

+

+=

++=

r

un

ixix

iyiyArci

ixixiyi

yid

122

,22

0

1

1

111

,120

1

1

==

==

QPK

m

Q

QPK

m

Q

0)nLnP2L2P1L1P(-x)nP2P1P( =++++++

( ) ( )nn2211-1

n21 LPLPLPPPP ++++++=x

nn Lxv

Lxv

Lxv

=

=

=

.....22

11

=

=

0

0

iy

ix

nv

nP

Tv

2v

2P

T2

v1

vl

PT1

vvll

PT

vn

+++=


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5. Mathematical Model

5.1 Determination of Data Source for the

Models

For leveling networks, the data used areobservations, height differences and the fixedheights. The unknown heights are determinedusing the height differences. In levelingnetworks, at least height of one point must beconsidered as fixed height. Weights arecalculated using the following formula;

pi=1/Si(km) (28)

For traverses, the data used are observations(angles, lengths), known coordinates andknown azimuths. In closed traverse, at leastone point and in open traverse, at least two

points must be selected as fixed points.Azimuths and lengths of other points aredetermined using the equation (37-38).Weights are determined by p=1/2 using thestandard deviations of the observations.Standard deviations of the angles should beconsidered as RADIAN in Excel calculations.

In satellite measurements, the data used are thecoordinates and cofactor matrix of the points.

5.2 Selection of Constrains for the Models

The constraints in Leveling adjustment;Loop Closure (Sum of the height difference) =

0The constraints in the closed traverseadjustment;Sum of x = 0; Sum of y = 0 Sum of interior angles = (n-2)*200 (for closed

traverse). Where n is the number of sides ofclosed traverse.Constraints in open traverses

(29)

(30)

Constraints in satellite measurements

(P1v1+P2v2++Pnvn) = 0 (31)

vTPllv = (L1TPlL1+ L2

TP2L2+.+ LnTPnLn) (32)

vTPllv = (L1TPlL1+ L2

TP2L2+.+ LnTPnLn) -

x

T

(P1L1+P2L2++PnLn) (33)

vTPllv=(L1TPlL1++ Ln

TPnLn)- (P1L1++PnLn)

T(P1+...+Pn)-1

(P1L1+P2L2++PnLn) (34)

5.3 Selection of Objective model

In the adjustment process, (vTPv) should beminimum since the goal of the adjustment is toobtain the value that has maximum probability.

6. The Some Adjustment Samples by UsingExcel Solver

6.1 Leveling Network

The leveling network example providedbelow is taken from El-Shimmy [5].

Fig. 10.Leveling Network

(35)

The weights of Observations are calculatedfrom formula (28). According to thisformula, six equations are written forweights of observations;

b

c

HA=0.00 m.

h4

h1

h6

h2

h5

h3d

a

iiiab

i

i

iiiab

i

i

dyyyy

dxxxx

sin0)(

cos0)(

2

1

2

1

==

==

=

=

chbhh

dh

bhh

ahbhh

dhchh

ahdhh

ahchh

6

5

4

3

2

1

+=

+=

=

+=

=

=


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(36)

Fig. 11.Adjustment of Leveling Networks

In figure 11; (B6:B11) contains heightsdifference, (C6:C11) contains levelingdistances, (E6:E11) contains weights and (D5)contains fixed height.Figure 11 shows the input for Solver. The cellsare contains following inputs;1. B15 contains the height differences in cellB6. The formula in B15 (=B6) is copiedthrough B20.2. C15 contains the height differences in cellC6. The formula in C15 (=CB6) is copiedthrough C20.3. D15 contains a formula for computing. Seeequation (36).D15(=H17-D5); D16(=H18-D5);D17(=-H17+H18); D18(=-D5+H16);D19 (= -H16+H18); D20 (= -H16+H17)4. E15 contains a formula for computing loopclosure. See equation (12). This formula isshown as belowE15(=H17-D5-B15); E16(=H18-D5-B16);E17(= -H17+H18-B17); E18(= -D5+H1-

B18);

E19(= -H16+H18-B19);E20(= -H16+H17-B20)5. F15 contains the weights in cell F6. Theformula in F15 (=F6) is copied through F20.6. G15 contains a formula for computing(VTPV).Where; v = Residuals, P = The weights ofobservationsG15(=F15*E5^2); G16(=F16*E5^2);G17(=F17*E5^2); G18(=F18*E5^2);

G19(=F19*E5^2);G20(=F20*E5^2)7. The cells H16, H17 and H18 contain aformula for computing point heights (hb, hcandhd). This formula is shown as belowH16 (=SUM (D5:B18)) ; H17 (=SUM(D5:15)); H18 (=SUM (D5:B16))8. G21 contains a formula for computing(vTPv). This formula is shown as belowG21 (=SUM (G15:G20))9. E23 contains a formula for computing loopclosure. E23 (=SUM (E15:E20))

km4

km2

km4

km2

km2

km4

07.5

58.11

09.1

41.6

57.12

16.6

6

5

43

2

1

1,6

(km.)Distancesection(m.)nsObservatio

==

h

h

h

h

h

h

l

S

25.041)(16

50.021)(15

25.041)(14

50.021)(13

50.021)(12

25.041)(11

===

===

===

===

===

===

kmSP

kmSP

kmSP

kmSP

kmSP

kmSP

=

dh

ch

bh

U

3,1 x

nkonown


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Performing of Leveling Adjustment in Excel

Solver

Select TOOLS/SOLVER from Excels

main menu. The SOLVER PARAMETERdialog box as shown in figure 12 below will bedisplayed.

Fig. 12. Solver Parameters Box

Set Target Cell: The target cell is G21. Thiscell contains the weighted sum of the squaresof the residual to be minimized.

Equal to: The Min radio button should beselected.

By Changing Cells: H16 to H18 are pointsheights which are calculated as approximationfrom height differences.Subject to the Constraints: Clicking ADD

button will display the ADD CONSTRAINTSdialog box in figure 13. For this example, thereis a constraint which is loop closure in cell

E23=0.

Fig. 13.Constraint Box

Solver Options: Clicking the options button inSOLVER PARAMETER dialog box willdisplay the SOLVER OPTIONS dialog box as

shown in figure 14. Leveling adjustment theoptions can be selected as shown in figure 14.

Fig. 14. Solver Options Box

Performing of Leveling Adjustment in Excel

The Adjustment can be performed byclicking SOLVE button in SOLVER

PARAMETER dialog box. If the adjustment issuccessful the Solver dialog box will bedisplayed as shown in figure 15. Click OK

button to save the solution results.

Adjusted Results

D15 to D20 contains corrections ofheight differences. E15 to E20 containscorrections of Loop closing. The cells H16,H17 and H18 contain the adjusted pointheights (hb, hc andhd ). The results are givenin figure 15 and 16.

Fig. 15. Adjustment of Leveling Networks bythe Excel SOLVER

Fig. 16.Summary Adjusted Results

6.2Traverse Connecting Two Fixed PointsThe traverse connecting two fixed points

is shown in figure 17. The points A, B, C, Dand the azimuths (AB), (CD) are fixed. All

bearings are direction clock wise. All distancesare measured from point A to point D.

Fig. 17.Traverse Connecting Two Fixed Points

Y= 8123.00 mX= 5787.87m

Y= 8228.84mX= 5058.48m

Y= 7988.10mX= 5121.87m

Y= 7909.20mX= 5104.71m

B

B=344g.3640

A

C

D

2

N

2=225g.7189

=185g.8790

1=139g.5761

1

dB1=152.37m.

d23=70.16m.

d12=44.26m.


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+

+=

++=

)1(

)1

(tan)(

)22

1()22

1(

ixix

i

y

i

y

Arci

ixixiyi

yid

(37)

(38)

n= i+ (39)

Where; di = The distance is between twopoints, i= azimuth

Settings Traverse Connecting Two Fixed

Points in Excel

Figure 18 show the observations in cell

(B6:B9, D6:D8) their standard deviations incell (E6:E8, G5:G10) and weights in cell(F6:F9, H6:H10), the fixed coordinates of

points in cell (L6:L9, M6:M9), the fixedazimuth in cell (I5 and J5).

The distances weights are calculated 1/(0.05)2=400g, and the angle weights are calculated(636619,7724/15,43209877)2 = 1701806811.

The cells (C5:C10) contains a formula forcomputing the azimuth from A to B. Seeequation (39).[=IF(C5+B6>600;B6+C5-600;IF(C5+B6>400;C5-200+B6;IF(C5+B6>200;B6+C5-200;B6+C5)))].See equation (40).

I6 contains fixed azimuth in grad unit. J6contains a formula for transformation fromGrad to Radian by the formula (=grd2r (I6)).

Fig. 18. Settings Traverse Connecting Two Fixed Points in Excel


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Table 1.Specifications of Traverse Connecting Two Fixed Points in Excel

Cell Number Formula Cell Number FormulaB17:B19 Observed Distances K19 =F8*H19^2

C17 =SQRT(M17^2+L17^2) K21 =SUM(K17:K19)C19 =SQRT(M19^2+L19^2) L17 =B17*SN(E17)D17 =grd2r(B6) L19 =B19*SN(E19)D20 =grd2r(B9) L21 =SUM(L17:L19)E16 =grd2r(C5) L24 =N21-L21E20 =grd2r(C9) M17 =B17*COS(E17)F16 =grd2r(C5) M19 =B19*COS(E19)F17 =grd2r(azmt(0;0;M17;L17)) M21 =SUM(M17:M19)F20 =grd2r(C10) M24 =O21-M21G17 =ANGLE(F16;F17) N16 =L7G20 =ANGLE(F19;F20) N17 =N16+L17H17 =G17-D17 N18 =N17+L18

H20 =G20-D20 N19 = N8I17 =C17-B17 N21 =N19-N16I19 =C19-B19 O16 =M7J17 =H6*H17^2 O17 =O16+M17J20 =H9*H20^2 O18 =O17+M18J21 =SUM(J17:J20) O19 =M8J24 =SUM(J21:K21) O19 =M8K17 =F6*H17^2 O21 =O19-O16

SOLVER Parameters

Fig. 19. Solver Parameters

Set Target Cell: Target cell is J24 and it iscontain the weighted sums of the squares of theresiduals in angles and distances to be

minimized.Equal to: The MN radio button should beselected.

By changing Cells:L17 to M19 are the Cartesiancoordinates differences (X, Y) that arecalculated from the observed distances and theazimuth of the lines derived from the observedangles. It also contains E20 contain calculatedazimuth (CD)Subject to the Constraints: In the traverseconnecting two fixed points are observed anglesand distances. For this example, number of the

residuals is equal to number of the constraints.

These constraints are sum of the coordinatedifferents in the direction of Y and X axis. Sum

of the coordinate differents are y= -0.08 m. andx = -0.06 m. fis closing error. It is calculatedby the formula f= (AB) + i (n*200). Fourangles and three distances are observed in thedirection of the clock wise. The unkonowns arecoordinates of point 1 and point 2. The numberof redundant observation (degrees of freedom) iscalculated by equation (18) and it is equal tothree.Clicking the ADD button will display the ADDCOSTRAINT dialog box. For this example,there are three constraints. These constraints are

given in below.1.The Azimuth that is calculated in the C point is

should be equal to the Azimuth that is fixed inthe C point. The constraint formula is E20 =F20.

2.Sum of the closing in the direction of y axis incell L24= 0.

3.Sum of the closing in the direction of x axis incell M24= 0.

Performing of Traverse Adjustment in Excel

The Adjustment can be performed by

clicking SOLVE button the options button in


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SOLVER PARAMETER dialog box. If theadjustment is successful the Solver dialog box

will be displayed as shown in figure 20. ClickOK button to save the solution results.

Fig. 20. Performing of Traverse Adjustment in Excel

Adjusted Results

The results of adjustment are given as summary in figure 21.

Fig. 21.Summary Adjusted Results

E33 to E36 contains corrections for observationangles. H17 (=rad2grd (H17))

contains a formula for transformation fromradian to grad (See appendix, [(=(=grd2rad(.)]).The formula in H17 is copied through H20.

E33 (=grd2rad (H17)) contains a formula fortransformation from grad to radian. The formulain E33 is copied through E36.F33 to F35 contains the residuals for distances.The formula in F33 (=I17) is copied throughF35.G33 to G36 contains the adjusted Azimuth. Thecells from G33 to G36 are transformed fromradian to grad by equation [(=rad2grd (F17))].H33 to H36 contains the adjusted distances. Thecell H33 (=C17) is copied through H35 (=C19).The cells (I34; J35) contain the adjusted

coordinates. The formula I34 (= N17) is copied

I35 (= N18) and J34 (= O17) is copied J35 (=O18)6.3Closed TraverseSettings Closed Traverse in Excel SOLVER

The cells (B7:B12, D7: D12) containobservations, the cells (E7:E12, G7:G12) containstandard derivation of observations and their

weights.The cells (L6, M6) contain fixed pointcoordinates.The distances weights are calculated 1/(0.05)2=400, and the angle weights are calculated(636619,7724/15,43209877)2 = 1701806811.The cells (C7:C12) contains a formula forcomputing the azimuth from A to B.[=IF(C5+B6>600;B6+C5-600;IF(C5+B6>400;C5-200+B6;IF(C5+B6>200;B6+C5-200;B6+C5)))].


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Fig. 22.Closed Traverse

Fig. 23. Settings Closed Traverse in Excel

Table 2. Specifications of Closed Traverse in Excel

Cell Number Formula Cell Number FormulaB23 =D7 G28 =C28-B28B28 = D12 J23 =H7*H23^2C23 =SQRT(L23^2+M23^2) J28 =H12*H28^2C28 =SQRT(L28^2+M28^2) J30 =SUM(J23:J28)

D23 =grd2r(B7) K23 =F7*H23^2D28 =grd2r(B12) K28 =F12*H28^2E22 =J6 K30 =SUM(K23:K28)E23 =grd2r(C7) L23 =B23*SN(E23)E28 =grd2r(C12) L28 =B28*SN(E28)F22 =J6 L30 =SUM(L23:L28)E23 =grd2r(azmt(0;0;M23;L23)) M23 =B23*COS(E23)E28 =grd2r(azmt(0;0;M28;L28)) M28 =B28*COS(E28)E29 =F22 M30 =SUM(M23:M28)F23 =ANGLE(F22;F23) N23 =N22+L23F28 =ANGLE(F27;F28) N28 =N27+L28H23 =G23-D23 O23 =O22+M23H28 =F29-F28 O28

G23 =C23-B23

4= 289.921g

P2

P1

P5

P4

P3

d34=59.49 m d45= 57.43 m

d51=47.10 m

d61= 37.91 m

P6

d23= 38.08 m

3= 205.359g

6=180.1854g

5=305.804g

1= 303.804gd12=75.73 m

2= 315.665g

(P1P2)=339 g.791

N


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SOLVER Parameters

Fig. 24. Solver parameters Dialog Box

Set Target Cell: Target cell is J30 and it is

contain the weighted sums of the squares of theresiduals in angles and distances to beminimized.

Equal to: The MIN radio button should beselected.

By changing Cells:L23 to M28 are the Cartesiancoordinates (X, Y) differences that are calculatedfrom the observed distances and the azimuth ofthe lines derived from the observed angles. Also,it contains the cell E28 that is calculatedazimuth.Subject to the Constraints:In the closed traverse

are observed angles and distances. For thisexample, Number of the residuals is equal tonumber of the constraints. These constraints aresum of the coordinate different in the directionof Y and X axis. Sum of the coordinate differentare y= -0.07 m. and x = -0.05 m. fis closingerror. It is calculated by the formula

f= (AB) + i (n*200). Six angles and sixdistances are observed in the direction of theclock wise. The unknowns are coordinates of

points P3, P4, P5 and P6. The number ofredundant observation (degrees of freedom) iscalculated by equation (18) and it is equal totwo.

Clicking the ADD button will display theADD COSTRAINT dialog box. For thisexample, there are three constraints. Theseconstraints are given in below.1.The Azimuth that is calculated in the P1 point

is should be equal to the Azimuth that is fixedin the P1 point. The constraint formula is E28=F29.

2.Sum of the closing in the direction of y axis incell L30= 0.

3.Sum of the closing in the direction of x axis incell M30= 0.

Performing of Traverse Adjustment by the Excel

Solver

The Adjustment can be performed byclicking SOLVE button in SOLVER

PARAMETER dialog box. If the adjustment issuccessful the Solver dialog box will bedisplayed as shown in figure 25. Click OK

button to save the solution results.

Fig. 25. Performing of Traverse Adjustment in Excel


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Adjusted Results

The results of adjustment are given as

summary in figure 26.

Fig. 26.Summarized Adjustment Results

B35 to B40 contains corrections for observationangles. C35 to C40 contains corrections forobservation distances. D35 to D40 containsadjusted azimuth. Also, D35 to D40 contains aformula for transformation from radian to grad

by the formula (=rad2grd (H17)).E35 to E40 contains adjusted distances. C45 toC51 contains adjusted coordinates in thedirection of Y axis. D45 to D51 containsadjusted coordinates in the direction of X axis.

6.4Adjustment of Coordinates Determinedfrom Satellite Observation

Geocentric coordinate values of a pointwere determined using Doppler Method. Thesatellite observations were collected in 4different sessions by taking advantage ofdifferent satellite constellations. A priori errors

of the coordinates are 1 meter.

Determining of Weight matrices

Variance- covariance matrices (Kll) arecalculated using the following formula,

(Kll)=m02(Qll) (40)

Where; (Qll) = Cofactor matrices, m0= a-priorivariance

For this problem, a-priori variance is equal to 1.Weight matrices are calculated using thefollowing formula,

Pll = Qll-1= Kll

-1 (41)

Where; Pll= Weight matrices

Determining of Approximate Coordinates

The Approximation coordinates arecalculated as to the smallest X, Y, and Zcoordinates that the satellite observations werecollected in 4 different sessions. The smallest Xcoordinate is at 4. period. The smallest Y and Zcoordinates is at 2. period. So, approximationcoordinates,X=3794214.00; Y=478192.00; Z=5172108.00

Settings for Adjustment of Coordinates

Determined from Satellite Observation in ExcelSOLVER

The cells (D21:D32) contain the coordinates thatis determined by Doppler method. The cells(F21:F32, G21:G32, H21:H32) contain cofactormatrices of the coordinates.The cells (E21, E32) contain the coordinatesdifferences.The weights are calculated. See equation (42).Excel supports several functions of a matrix. Foradditional information the reader is encouragedto visit the references (8-9). The referencescontains to the most current information on theMatrix solution.


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Fig. 27. Settings for Adjustment of Coordinates Determined from Satellite Observation in Excel

Table 3.Specifications of Coordinates Determined from Satellite Observation in Excel

CellNumber

FormulaCell

NumberFormula

D21 =D5 F36 =SUM(J22;J25;J28;J31)D32 =D16 F37 =SUM(J23;J26;J29;J32)F21 =E5 G35 =SUM(K21;K24;K27;K30)F32 =E16 G36 =SUM(K22;K25;K28;K31)G21 =F5 G37 =SUM(K23;K26;K29;K32)G32 =F16 H35 =INDICE(MINVERSE (E35:G37);1;1)H21 =G5 H36 =INDICE(MINVERSE (E35:G37);2;1)H32 =G16 H37 =INDICE(MINVERSE(E35:G37);3;1)

E21 =D21-D35 I35 =INDICE(MINVERSE(E35:G37);1;2)E22 =D22-D36 I36 =INDICE(MINVERSE(E35:G37);2;2)E23 =D23-D37 I37 =INDICE(MINVERSE(E35:G37);3;2)E24 =D24-D35 J35 =INDICE(MINVERSE(E35:G37);1;3)E25 =D25-D36 J36 =INDICE(MINVERSE(E35:G37);2;3)E26 =D26-D37 J37 =INDICE(MINVERSE(E35:G37);3;3)E27 =D27-D35 C41 =INDICE(MMULT(I21:K23;E21:E23);1;1)E28 =D28-D36 C42 =INDICE(MMULT(I21:K23;E21:E23);2;1)E29 =D29-D37 C43 =INDICE(MMULT(I21:K23;E21:E23);3;1)E30 =D30-D35 D41 =INDICE(MMULT(I24:K26;E24:E26);1;1)E31 =D31-D36 D42 =INDICE(MMULT(I24:K26;E24:E26);2;1)E32 =D32-D37 D43 =INDICE(MMULT(I24:K26;E24:E26);3;1)I21 =INDICE(MINVERSE(F21:H23);1;1) E41 =INDICE(MMULT(I27:K29;E27:E29);1;1)I32 =INDICE(MINVERSE(F30:H32);3;1) E42 =INDICE(MMULT(I27:K29;E27:E29);2;1)J21 =INDICE(MINVERSE(F21:H23);1;2) E43 =INDICE(MMULT(I27:K29;E27:E29);3;1)J32 =INDICE(MINVERSE(F30:H32);3;2) F41 =INDICE(MMULT(I30:K32;E30:E32);1;1)K21 =INDICE(MINVERSE(F21:H23);1;3) F42 =INDICE(MMULT(I30:K32;E30:E32);2;1)K32 =INDICE(MINVERSE(F30:H32);3;3) F43 =INDICE(MMULT(I30:K32;E30:E32);3;1)D35 Approximation X coordinate H41 =SUM(C41:F41)D36 Approximation Y coordinate H42 =SUM(C42:F42)D37 Approximation Z coordinate H43 =SUM(C43:F43)E35 =SUM(I21;I24;I27;I30) J41 =INDICE(MMULT(H35:J37;H41:H43);1;1)E36 =SUM(I22;I25;I28;I31) J42 =INDICE(MMULT(H35:J37;H41:H43);2;1)E37 =SUM(I23;I26;I29;I32) J43 =INDICE(MMULT(H35:J37;H41:H43);3;1)F35 =SUM(J21;J24;J27;J30)


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7. ResultsIn this study, Standard Excel Solver that

the software comes with Microsoft Office CD-

Rom. is used. Standard Excel Solver supportsjust 200 decision variables [4]. The adjustment ofmedium and small geodetic networks withstandard Excel Solver is done easiness. ThePremium modules of Excel Solver for adjustmentof bigger networks must used. Premium ExcelSolver supports approximately 32000 decisionvariables [4]. Premium modules requireadditional cost.

It has been proved that Excel Solver can beused to adjust 3D coordinates of the points

pertaining to leveling networks, closed and open

traverses using least squares method. Excelsolver also can be used to adjust the 3Dcoordinates of the points determined by GPSobservations. The results obtained from ExcelSolver are exactly the same as the resultsobtained from conventional data adjustmenttools.Excel Solver has the following advantage anddisadvantages;1.It is not necessary to have a broad knowledge

about least squares technique if Excel Solver isused. It will be enough if the user has

fundamental knowledge about data adjustment.2.It is very important to select the cells to be

minimized and chosen as constraints correctly.3.The solution converges quickly.4.Excel Solver gives flexibility and simplicity to

the user in the selection of parameters andconstraints.

5.Standard Excel Solver does not requireadditional cost, for the software comes withMicrosoft Office CD-Rom.

6.It does not require complex codes to performdata adjustment.

7.Users dont need complex adjustmentprograms for adjustment of geodetic networks.

8.Necessary statistical knowledge for analysis ofadjustment results arent obtaining from ExcelSolver.

8. References1. ztrk E., 1991,Dengeleme HesabCilt 1, K.T.

Mhendislik Mimarlk Fakltesi, Yayn No: 119,Trabzon.

2. Hashimi S, R., 2004, Traverse Adjustment UsingMicrosoft Excel Solver, ACSM/TAPSConference, April 19-21, Nashville,TN.

3. http://www.microsoft.com, November 4, 2004,How to Create Visual Basic Macros by Using

Excel Solver in Excel 97, Article ID: 8433044. Fylstra, D., Lasdon, L., Watson, J., Waren, A.,

1998, Design and Use of the Microsoft ExcelSolver. Institute for Operations Research andManagement Sciences.

5. El-Sheimy, N., Lecture notes Adjustment ofObservation, ENGO 362, The University ofCalgary Department of Geomatics Engineering.Canada.

6. http://hspm.sph.sc.edu/Courses/J716/SolverInstall.html, January 13, 2004

7. Ziggy M., 1995, Teaching Linear Programmingusing Microsoft Excel Solver,Cheer (Computersin Higher Education Economics Rewiev),Volume 9, Issue 3.

8. Veinott, A, F., 2004, Formulating and SolvingLinear Programs in Excel Solver, Introduction toOptimization, MS&E 111.

9. Roy, B, V., 2003, Excel Solver Tutorial,ENGR62/MS&E 111.

the adjustment of some geodetic networks using microsoft excel

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