The Two Peg TestThe Two Peg Test

Reciprocal MethodReciprocal Method

To Determine the collimation error of a Level InstrumentTo Determine the collimation error of a Level Instrument

A few reminders: see levelling presentation

We will use convention that error is the result of the observed value minus the true value

It follows that the correction to the observed value then must have the opposite sign as the error

RR = Rod Reading

BS = Backsight RR

FS = Foresight RR

EL = ElevationEL Elevation

DEL = difference in elevation = BS – FS ( one setup) = ΣBS – ΣFS (involving a number of turns)

DEL = final EL – initial EL

H H i t l Di tH = Horizontal Distance

Σ = summation sum of e.g. ΣBS = BS1 + BS2 + BS3 . . . + BSLast

The Level Instrument 2-Peg TestThe Instrument set up andset up and levelled

Horizontal Line

A

B

Neglecting curvature and refraction

in 60 m c+r =0.0002m

d

B

dL

Reciprocal MethodSet up as near to the A position as focus will allow

1st setup near AA RR=aed = RR due to collimation error = RR with error =ad+ed

Set up as near to the A position as focus will allow

ed = Collimation error over distance daed

Read and book BS on A

ad What proper reading should beed = Collimation error over distance d

A

B

Place 2 pegs A & B about 60m is a reasonable Horizontal distance apart

d

B

dMinimum focus distance

L

1st setup near AA RR=aedAB

RR aed

aed-be(d+L)

=(ad+ed)-(b(d+L)+ed+L)

RR=be(d+L) =RR with collimation error

R d d b k

ed+LCollimation error over distance d+L =

(ad ed) (b(d+L) ed+L)

be(d+L)

Read and book FS on B

b(d+L)What proper reading should be >

A

B

d

B

dL

1st setup near AA RR=aedAB

RR aed

RR=be(d+L)

aed-be(d+L)

=RR with collimation error

=(ad+ed)-(b(d+L)+ed+L)

be(d+L)

ed+L

(ad ed) (b(d+L) ed+L)=ad-bd+L+ed-ed+L = DELappAB1 = (DELAB)-eL

eLaed

b(d+L)eded Horizontal Linesad

A

BDELAB=EL diff. A to B = ELB-ELA

( ) (b )DELAB

d

B=(ad) - (bd+L)

Elevation B

Elevation A

dL

Datum(0)

Elevation B

1st setup near A 2nd setup Near Peg BA RR=aedAB

RR aed

RR=be(d+L)

aed-be(d+L)

=(ad+ed)-(b(d+L)+ed+L)

RR=bed

For second setup do so as near to B as

( ) ( ( ) )

bed

bded

near to B as focus will allowRead and book

A

BB

L d

1st setup near A 2nd setup Near Peg BA RR=aed RR=ae(d+L)AB

RR aed

RR=be(d+L)

aed-be(d+L)

=(ad+ed)-(b(d+L)+ed+L)

RR=bed

RR=ae(d+L)

ae(d+L) - bed

=(a(d+L)+ed+L)-(bd+ed)

Read and book

( ) ( ( ) ) ( ( ) ) ( )

ae(d+L)

a(d+L)

ed+L

Read and book

A

BB

L d

1st setup near A 2nd setup Near Peg BA RR=aed RR=ae(d+L)AB

RR aed

RR=be(d+L)

aed-be(d+L)

=(ad+ed)-(b(d+L)+ed+L)

RR=bed

RR ae(d+L)

ae(d+L) - bed

=(a(d+L)+ed+L)-(bd+ed)

Read and book

( ) ( ( ) )=a(d+L)-bd+ed+L-ed = DELappAB2 = (DELAB)+eL

( ( ) ) ( )

ae(d+L)

a(d+L)

ed+L

Read and book

eded

eL

bd

bed

A

B

eded

DELAB=EL diff. A to B = ELB-ELA

=(a(d+L))-(bd)

bd

B

L d

aed-be(d+L) =(ad+ed)-(b(d+L)+ed+L) =ad-bd+L+ed-ed+L = DELappAB1 = (DELAB)-eL1st

2nd ae(d+L) - bed =(a(d+L)+ed+L)-(bd+ed) =a(d+L)-bd+ed+L-ed = DELappAB2 = (DELAB)+eL

Adding the 1st result to 2nd leaves 2*(DELAB)

Therefore dividing by 2 leaves DELABTherefore dividing by 2 leaves DELAB

Or Subtracting the 1st result from the 2nd leaves 2* eL

Therefore dividing by 2 leaves eL The collimation error over dist. L

Note: Even if d in the 1st setup and the 2nd setup were not equal and then ed for the 1st setup would not equal ed for the 2nd setup, they would still cancel for each setup

A

BDELAB

B

Elevation B

Elevation A

L

Elevation B

Datum(0)

eL = collimation error in Horiz. Dist. L \ eL/L= collimation error per unit length = e1

Then –e1= correction to rod reading per unit of Horiz. Dist. = c

ed+L

RR including Collim. Errorbed+L

eLed

aed =RR with error

bd+LWhat rod reading should bead =what RR should be

AThus we could correct a RR for any Horiz. Dist. (H)

H*c = correction to RR

d

BThe error over d+L (ed+l) = error over d (ed) + error over L (eL)

(ed+L)=(ed)+(eL)dL

(ed+L)=(ed)+(eL)d + L

Reciprocal Method Example L=60m, d=2mSet up as near to the A position as focus will allowSet up as near to the A position as focus will allow

1st setup near ARR=1.900 = ad + edA

aed=1.900ed = Collimation error over distance d

Read and book BS on A

ed

ad What proper reading should be= Collimation error over distance d

A

B

d

B

dL

Minimum focus distance

1st setup near ARR=1 900 = a + e = a +eA

R d d b k

RR=1.900 = ad + ed = ad +ed

RR=2.456= bd+L + ed+L = bd+L + ed + eL

1.900-2.456= -0.556

AB

= ad+ed-bd+L-ed-eL = DELapp1 = DEL-eL

be(d+L)=2.456

ed+L

Read and book FS on B

eLaed

b(d+L)eded Horizontal Linesad

A

B

2m

B

2m60m

Set up as near to the B position as focus will allow2nd setup near B

ARR=1.800 = bd + edB

A

bed=1.800bded

Read and book

A

BB

60m 2m

2nd setup near BRR=1.256= a(d+L) + ed+L = ad+L + ed + eL

RR=1.800= bd + ed = bd + ed

1.256-1.800= -0.544

AB

= ad+L + ed + eL - bd - ed = DELapp2 = DEL+eL

Read and bookae(d+L)=1.256

a(d+L)

ed+L

Read and book

eded

eL

bd

bed=1.800

A

B

eded

DELAB=EL diff. A to B = ELB-ELA

=(a(d+L))-(bd)

bd

B

60m 2m

aed-be(d+L)=1.900-2.456= -0.556 =(ad+ed)-(b(d+L)+ed+L) =ad-b(d+L)+ed-ed+L =DELappAB1=(DELAB)-eL1st

2nd ae(d+L)-bed=1 256-1 800= -0 544 =(a(d+L)+ed+L)-(bd+ed) =a(d L) bd+ed L ed =DEL AB2=(DELAB)+eL2 d ae(d+L)-bed=1.256-1.800= -0.544 =(a(d+L)+ed+L)-(bd+ed) =a(d+L)-bd+ed+L-ed =DELappAB2=(DELAB)+eL

:Adding the 1st result to 2nd leaves 2*(DELAB)

:Therefore then dividing by 2 leaves DELAB

-1.100

-0.550

:Or Subtracting the 1st result from the 2nd leaves 2* eL

:Therefore then dividing by 2 leaves eL

Th lli ti di t L 60

0.012

0.006 The collimation error over dist. L=60m

Note: Even if d in the 1st setup and the 2nd setup were not equal and then ed for the 1st setup would not equal ed for the 2nd setup, they would still cancel for each setup

0.006

A

BDELAB

B

Elevation B

Elevation A

60m

Elevation B

Datum(0)

eL = collimation error in Horiz. Dist. L = 0.006m\ eL/L= collimation error per unit length = e1=0.006m/60m=0.0001m/mTh ti t d di it f H i Di t 0 0001 /Then –e1= correction to rod reading per unit of Horiz. Dist. = c = -0.0001m/m

Thus we could correct a RR for any Horiz. Dist. (H)

H*c = correction to RR

RR including Collim Error

H*c = correction to RR

ae(d+L)

a(d+L)

ed+L

eded

eL

bd

RR with error = bed

Collim. Error

Wh t dA

B

eded bdWhat rod reading should be

What rod reading should be

B

The error over d+L:(ed+l) = error over d:(ed) + error over L:(eL)(ed+L)=(ed)+(eL)

L d

(ed+L)=(ed)+(eL)

c = -0.0001m/m = Collimation Factor = c factor

H*c = correction to RR

Ex. For previous 2nd setup reading to A : H=L+d=62m

H*c = correction to RR = 62*(-.0001) = -0.0062ma ( )=1 256 1 256+( 0 006) = 1 250 = a( )

RR including Collim Error

ae(d+L)=1.256 1.256+(-0.006) = 1.250 = a(d+L)

What the correct RR should have been

If adjusting instrument this is the

ae(d+L)

a(d+L)

ed+L

eded

eL

bd

RR with error = bed

Collim. Error

Wh t d

reading you would adjust +hair to

A

B

eded bdWhat rod reading should be

What rod reading should be

B

The error over d+L:(ed+l) = error over d:(ed) + error over L:(eL)(ed+L)=(ed)+(eL)

60m 2m(ed+L)=(ed)+(eL)

A

B

d

B

dL

A

B

d

B

dL

Corrections involved in 1 setup

HF*c=cHF

HG*c=cHG

Corrected RR @ F = f = fe + cHF = fe+HF*cCorrected RR @ F f fe cHF fe HF cCorrected RR @ G = g = ge + cHG = ge+HG*c

DEL from F to G = f - g = fe - ge + (HF – HG)*cRemember DEL = BS - FS

f g

g g ( )

BSFS

Correction @ HF=cHF f

fe

g

ge

Correction @

F@ HG=cHG

G

HF

HHF+HG

HG

DEL from F to G = f – g = fe – ge + (HF – HG)*c

Then in general For any setup

DEL = BScorrected – FScorrected = BSobserved – FSobserved + (Hto BS – Hto FS)*c

DEL = BScorrected – FScorrected = BSobserved – FSobserved + (imbalance)*c

For any section involving a number of turns (since linear)

f gBS

FS

For any section involving a number of turns (since linear)

DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved–ΣFSobserved+(ΣHto BS – ΣHto FS)*c

DEL = ΣBScorrected–ΣFScorrected=ΣBSobserved –ΣFSobserved+(total imbalance)*c

Correction @ HF=cHF

Correction @

f

fe

g

ge

F@ HG=cHG

G

HF

HHF+HG

HG

Correction @ HF=cHF

Correction @

f

F@ HG=cHG

G

HF

HHF+HG

HG

Example HC=30.000m HD=70.000m

C

D

HC

HHC+HD

HD

A

B

H1+H2