06 traversecomputations w

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ERE 371

Traverse Computations

• Purpose

• Approaches to adjustment



ERE 371

Compass Rule Adjustment

• Procedure



ERE 371

Example Traverse

65° 33' 3!

""° 5' !

"""° #5' 3!

$#° 5%' 3! A

&

C

3 ( )5 $ 6 m

# 5 ) 6 5 m

5()%3$ m

3

) " % "

m



ERE 371

&alancin* An*les

• Purpose

• Procedure



ERE 371

+ethods ,or Adjustin* An*les

• -everal possi.ilities

• -election o, method



ERE 371

E/ual Corrections

• 0ssue

• istri.ute closure

-tation +easured An*le Correction Adjusted An*le

A 65° 33' 3! ! 65° 33' 3!

& ""° 5' ! 3! ""° 5' 3!

C """° #5' 3! 3! """° #5' !

$#° 5%' 3! 3! $#° 55' !

-1+ 352° 5(' 3! "' 3! 36° ' !



ERE 371

E/ual Corrections to An*les

• Advanta*es



ERE 371

Example Traverse &alanced An*les

65° 33' 3!

""° 5' 3!

"""° #6' !

$#° 55' ! A

&

C

3 ( )5 $ 6 m

# 5 ) 6 5

m

5()%3$ m

3

) " % "

m



ERE 371

irections o, 4ines

• 1se adjusted an*les to compute line direction

• Aimuths or .earin*s



ERE 371

Computin* irections

""° 5' 3!

A

&

d i r e c t i o n

3 °

" 5 ' 3

! C



ERE 371

Compute irections Cloc78ise Around Traverse

""° 5' 3!

A

&

" ° " ' !

3 °

" 5 ' 3 !

C

Assume or ,ind aimuth o, A&

3° "5' 3!

Compute aimuth o, &A 9aimuth o, A& ± "(°:

#"° "5' 3!

-u.tract an*le at &

#"° "5' 3! ; ""° 5' 3! < "° "' !



ERE 371

irections o, Traverse 4ines

-tation 4ine An*le Aimuth &earin*

A& 3° "5' 3! = 3° "5' 3! E

& ""° 5' 3!

&C "° "' ! - $2° 5' ! EC """° #6' !

C "6(° %%' ! - ""° "6' ! E

$#° 5%' 3!

A #$5° %2' ! = (%° ""' ! > A 65° 33' 3!

A& 3° "5' 3! = 3° "5' 3! E



ERE 371

irections o, Traverse 4ines

A

&

C

" ° " ' !

" 6 ( ° %

% ' !

#$ 5° %2' !

3 °

" 5 ' 3

!



ERE 371

epartures and 4atitudes

• epartures 9deps:

• 4atitudes 9lats:



ERE 371

Computin* epartures ? 4atitudes

• Compute .@ ep < 4 sin α 4at < 4 cos α

• >here α < aimuth 4 < len*th o, line

East 9B:

D

ep) D < 4 sinα

4at) D < 4 cosα α4

=orth 9:



ERE 371

Example Traverse

A

&

C

3 ( )5 $ 6 m

3

) " % "

m

#$ 5° %2' !

3 °

" 5 ' 3

!

" ° " ' !

" 6 ( ° %

%

' !

# 5 ) 6

5 m

5()%3$ m



ERE 371

Example Traverse

eparture < 4en*th x sin 9aimuth:e)*) ep) A& < 3)"%" m x sin93° "5' 3!:

4atitude < 4en*th x cos 9aimuth:

e)*) 4at) A& < 3)"%" m x cos 93° "5' 3!:

4ine istance Aimuth eparture 4atitude

A& 3)"%" m 3° "5' 3! "5)"(( m #6)3%$ m

&C 3()5$6 m "° "' ! 3$)2$3 m ;6)(2" m

C #5)65 m "6(° %%' ! 5)#6 m ;#5)"""6 m

A 5()%3$ m #$5° %2' ! ;5()"36" m 5)2##3 m

-um "5#)$52 m )#%( m )363 m



ERE 371

Closure in eps and 4ats

• or mathematicall@ closed traverse

• Deometricall@ closed traverse

• Closure di,,erence .et8een 7no8nFcomputed position

G 4inear error o, closure 94EC:

G Relative error o, closure 9REC:



ERE 371

4inear Error o, Closure

A

A'

Σdep <)#%( m

Σlat <)363 m

m)%%9)363:9)#%(:4EC e)*)

lat:9dep:94EC##

##

=+=

Σ+Σ=

)%% m

3%$

"

3%$")(

"

m"5#)$52

m)%%

perimeter

4ECREC ====



ERE 371

Traverse Adjustment

• Doal

• -ome methods

G Ar.itrar@ method

G Compass 9&o8ditch: rule

G 4east s/uares adjustment



ERE 371

Readin*s

• Chapter " G sections ")" G ")(



ERE 371

Compass Rule Adjustment

• Application

• >or7s ,or traverses 8ith limited num.er o, lines

len*th traverse

len*th line

closure lat

correction lat=

len*th traverse

len*th line

closure dep

correction dep=



ERE 371

Compass Rule

• Proportion is rearran*ed ,or computational e,,icienc@

len*th line

len*th traverse

closure latcorrection lat ×=

len*th linelen*th traverse

closure depcorrection dep ×=



ERE 371

Compass Rule G &alance epartures

len*th)"6#35len*thm"5#)$52

m:9)#%(; corr dep)

len*thperimeter

closuredeparturecorrection'eparture

×=×=∴

×=

4ine istance epartureepartureCorrection

Correctedeparture

A& 3)"%" m "5)"(( m ;)%2 m "5)"(3 m

&C 3()5$6 m 3$)2$3 m ;)63 m 3$)26% mC #5)65 m 5)#6 m ;)%# m %)22( m

A 5()%3$ m ;5()"36" m ;)25 m ;5()"%5 m

-um "5#)$52 m )#%( m ;)#%( m ) m

Corrected departure < eparture eparture correctione)*) Corrected departure A& < "5)"(( 9;)%2: < "5)"(3 m



ERE 371

Compass Rule G &alance 4atitudes

len*th)#3$63len*thm"5#)$52

9)363:; corr lat)

len*thperimeter

closurelatitudecorrection4atitude

×=×=∴

×=

Corrected latitude < 4atitude 4atitude correction

e)*) Corrected latitude A& < #6)3%$ 9;)$": < #6)#( m

4ine istance 4atitude4atitude

CorrectionCorrected4atitude

A& 3)"%" m #6)3%$ m ;)$" m #6)#( m

&C 3()5$6 m ;6)(2" m ;)2# m ;6)("( mC #5)65 m ;#5)"""6 m ;)6" m ;#5)""( m

A 5()%3$ m 5)2##3 m ;)"32 m 5)2( m

-um "5#)$52 m )363 m ;)363 m ) m



ERE 371

Computation o, Coordinates

• Deneral

• Application



ERE 371

Calculate Coordinates

m53#)23#)25 A&4atitude EE""()$"m"()$"" A&'epartureBB

A&

A&

=+=+==+=+=

-tation 4ine eparture 4atitude B Coordinate Coordinate

A ") m 5) m

A& "5)"(3 m #6)#$ m

& ""5)"(3 m 5#6)#$ m

&C 3$)26% m ;6)("( m

C "53)"%$ m 5"2)#2 mC %)22( m ;#5)""( m

"5()"%6 m %22%)2# m

A ;5()"%6 m 5)2( m

A ") m 5) m

m5"()("3)#253#)2&C4atitude EE

"5%)(2$m36)"26""()$"&C'epartureBB

CC

&C

=−=+==+=+=



ERE 371

Adjusted Aimuths and 4en*ths

D

ep) D

4at) D

East 9B:

=orth 9:

4

4 < len*th o, line D

## latdep4 +=

= −

lat

deptan "α

α

α < aimuth o, line D



ERE 371

Calculatin* Aimuths and 4en*ths

East 9B:H

I

4at) HI

ep) HI

4 < len*th o, line HI

4

=orth 9: ## latdep4 +=

= −

lat

deptan "α

α

α < aimuth o, line HI



ERE 371

Compute Adjusted AimuthsF4en*ths

-tation 4ine istance Aimuth B Coordinate Coordinate

A ") m 5) m

A& 3)"3# m 3° "5' #5!

& ""5)"(3 m 5#6)#$ m&C 3()5$" m "° "' 55!

C "53)"%$ m 5"2)#2 m

C #5)6" m "6(° %%' %!

"5()"%6 m %22%)2# m

A 5()%%5 m #$5° %(' "!

A ") m 5) m



ERE 371

4east -/uares Adjustment

• Theor@

• Application

• 0ssue



ERE 371

1nderstandin* 4-

• Advanta*es o, 4- approach

• Assumptions



ERE 371

1nderstandin* 4-

• Denerall@ ma7e redundant measurements

• Residual

G i,,erence .et8een o.servation and +PJ

e)*) νi < +PJ G o.si

• Premise o, 4-

minimum#m

#3

##

#"

m

"i

#i =++++=∑

=

ν ν ν ν ν



ERE 371

Example

• +easure distance ,ive times

G ")%K ")"5K 22)2$K 22)2%K ")6

• Ever@ o.servation has associated residual

G + < ")% ν" ∴ ν" < + G ")%

G + < ")"5 ν# ∴ ν# < + G ")"5

G + < 22)2$ ν3 ∴ ν3 < + G 22)2$

G + < 22)2% ν% ∴ ν% < + G 22)2%

G + < ")6 ν5 ∴ ν5 < + G ")6

• 4- selects + to minimie sum o, s/uared residuals

minimum#

5

#

%

#

3

#

#

#

"

5

"i

#

i =++++=∑=

ν ν ν ν ν ν



Example cont)

• >ant to minimie sum o, residuals

######i ")6:9+22)2%:9+22)2$:9+")"5:9+")%:9+L −+−+−+−+−=∑

")6:#9+22)2%:#9+22)2$:#9+")"5:#9+")%:#9++

L#

i =−+−+−+−+−=∂∂∑

minimum#

5

#

%

#

3

#

#

#

"

#

i =++++=∑ ν ν ν ν ν ν

"")%"+ =− ")"+ =∴

• Recall ,rom calculus

G unction minimum occurs 8hen derivative is ero

∴Ta7e derivate 8ith respect to + and e/uate to ero



ERE 371

4- Adjustment o, Traverse

• Components G +easure an*les and distances

G erive +PJ o, point coordinates

• M.servation e/uations

G Express o.served measurements in terms o, coordinates

#

A&

#

A& A& A& :9:B9B4 −+−=+υ

'

BB

tan

BB

tan A A &C

&C"

&C

&C"

&A&C A&C A&C +

−

−

−

−

−

=−=+ −−

υ θ

• 0ssue

G E/uations are non;linear

G =ormalied throu*h Ta@lor expansion



ERE 371

Noriontal Traverse 4- Adjustment

• +inimal input



ERE 371

Estimatin* Noriontal Coordinates

• Approach

-tation 4ine eparture 4atitude Estimated B Estimated

A ") m 5) m

A& "5)"(( m #6)35 m

& ""5)"(( m 5#6)35 m

&C 3$)2$ m ;6)(2 m

C "53)"5( m 5"2)##6 mC 5)3 m ;#5)""# m

"5()"6" m %22%)""% m

A ;5()"36 m 5)2## m



ERE 371

0nput ile ,or 4- Adjustment

Title

OdistancesK Oan*lesK OaimuthsK

Ocontrol ptsK total O pts

Ino8n control point 0K BK

1n7no8n points 0K est) BK est)

istance ,romK toK distance

An*les &-K 0nstrK -K de* min sec

Aimuth 0nstrK -K de* min sec



ERE 371

Mutput ile ,or 4- Adjustment

• Contents o, output ,ile

G -ummar@ o, input data

G Adjusted stations

G Adjusted distances and residuals

G Adjusted an*les and residuals

G Adjusted aimuths and residuals

G Associated statistics

• ualit@ o, adjustment G -ie o, all residuals

G Consider - o, unit 8ei*ht



ERE 371

Mutput rom 4- Adjustment

-tation 4ine istance Aimuth B Coordinate Coordinate

A ") m 5) m

A& 3)"#$ m 3° "5' #5!

& ""5)"(" m 5#6)#3 m

&C 3()562 m "° "' 55!

C "53)"%3 m 5"2)#5 m

C #5)6"% m "6(° %%' #5!

"5()"%% m %22%)(% m

A 5()%%5 m #$5° %(' 35!

A ") m 5) m



Readin*s

• Chapter " G sections ")""K ")"6 G ")"$

• Chapter "5 G sections "5)" G "5)3

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