SURVEY ADJUSTMENTS. UNIT III SURVEY ADJUSTMENTS CONTENTS Errors Sources, precautions and corrections...

SURVEY ADJUSTMENTS

2

UNIT III SURVEY ADJUSTMENTS

CONTENTS• Errors • Sources, precautions and corrections• Classification of errors • True and most probable values • Weighted observations • Method of equal shifts • Principle of least squares • Normal equation • Correlates • Level nets• Adjustment of simple triangulation networks.

Sivapriya Vijayasimhan

3


ErrorsMeasurement and Uncertainty• Measurements are rarely exactly the same• Measurements are always some what different from the “true value” • These deviations from the true value are called errors

Sources of ErrorsTwo sources of error in a measurement are • limitations in the sensitivity of the instruments • imperfections in experimental design or measurement techniques

Errors are often classified as: • Mistakes • Systematic • Accidental


4


Types of Error Sources Precautions

Mistakes -Ignorance, inexperience or carelessness - Poor judgement- Incorrect settings in equipment - Setup over wrong point

- Close traverse System - Independent field observation

Systematic - Typically present - Follows mathematical or physical law- Instrumental, physical and human limitations- Effect is cumulative Example: Device is out-of calibration

- Careful calibration. - Best possible techniques.

Accidental - Remain after eliminating mistakes and systematic error- Obey law of chance - Changes in experimental conditions

- Take repeated measurements and calculate their average.


5


QuantitiesIndependent Quantity : Independent of values of other quantities .Example: Reduced levelConditioned Quantity : Dependent upon the values of one or more quantity. Also called as

dependent quantityObserved value of quantity: Value obtained after all correction of errorsTrue Value of Quantity: Absolutely free from all errors .It is indeterminate and it is never

known

ObservationsDirect Observation: Measured directly upon desired quantity Eg. Measurement of single angleIndirect Observation: Observed value is deduced from measurement of some related

quantities. Eg. Measurement of a summation angle for the sum of an angle by repetitionWeight of an Observation : Number giving an indication of its precision and trustworthiness

when making comparison between several quantities of different worthWeightage = 5 (5 times as much as an observation of weight 1)

Weighted Observation: Different weights assigned to them. Unequal care and dissimilar condition exist at time of observation Arbitrarily


0180 CBA

6


Most Probable Value: It is the quantity of which has more chances of being true than has many other. Deduced from several measurements or from which it is based

Proved from theory of error 1. It is equal to arithmetic mean if observations are equal weight 2. It is equal to the weighted arithmetic mean in caseof observations of unequal weightsMost Probable Error: Quantity which is added to and subtracted from the most probable value

which fixes the limitTrue Error: Difference between the true value of an quantity and its observed valueResidual Error :Difference between most probable value of a quantity and its observed value

EquationObservation Equation: Relation between observed quantity and its numerical valueConditioned Equation: Relation between several existing dependant quantities Normal Equation: Formed by multiplying each equation by the c0-efficient of the un-known

whose normal equation is to be found and by adding thus formed equation.


7


Law of Accidental Errors Law of probabilityErrors in form of equation which is used to compute the probabale values or precision of a

quantity

Features1. Small errors tend to be more frequent than large one2. +ve and –ve of same size happen with equal frequency 3. Large errors occur impossible

Probable error of single measurement


Size of error

1

Σ6745.0

2

n

vEs

Difference in any single observation and mean of series

No. Of observation in series

8



n

EE s

m =

223

22

21 nEEEE +++

Probable error of an average,

Probable error of sum,

Where, E12 , E2

2, E32, En

2 are probable errors

PRINCIPLES OF LEAST SQUAREIn observations of equal precision the most probable values of the observed quantities are those that render the sum of squares of the residual errors a minimum

V1,V2,….Vn be the observed valuesX- most probable valueE1,E2…..En be the respective error of the observed valuesM – arithmetic valueN- number of observation

nn

n

EVX

EVX

EVXn

V

n

VVVM

'

..

22

11

21

9



MXn

EM

n

E

n

VX

EVnX

E - small and if more observations are made ΣV/n becomes negligible

1. Arithmetic mean is the true value where the number of observed values is less and the measurements are precise

R1,R2…Rn be the residual (difference between mean value and observed value)

n

VM

n

R

n

VM

RVnM

RVM

RVM

RVM

nn

'22

11

(ΣR/n = 0)

--------------1

10


2. The sum of residual equals zero and the sum of plus residuals equals the sum of minus residuals

N – any other unknown value other than arithmetic mean

Square the equation 1 and 2

Substitute equation 5 in 4


'

'22

'11

'

nn RVN

RVN

RVN

------------------------2

VMVVMVMVR

VNVnNR

VMVnMR

222

222'

222

2

2

2

n

VV

22

n

VRV

222

Substitute nM=ΣV

Substitute M=ΣV/n

------------------------4

------------------------3

---------------------------------5

22

2222' 2

n

VNnRVN

n

VRnVR

11


3. Sum of squares of the residual errors found by the use of the arithmetic mean is a minimum

LAWS OF WEIGHTS

1. Weights of the arithmetic mean of observations of unit weight is equal to the number of observations

Let P be measured angle for 4 times

Arithmetic mean = 35020’ +1 (10”+ 12”+ 8”+ 10”) = 35020’ 10”

4Weight of arithmetic mean = Number of observations = 4


Angle , P Weight

35020’10” 1

35020’12” 1

35020’8” 1

35020’10” 1

12


2.Weight of the weighted arithmetic mean is equal to the sum of the individual weights

Sum of individual weights = 2+3+4+2=11Weighted arithmetic mean = 35020’ + 1 (10”+ 12”+ 8”+ 10”) = 35020’3.64” 11Weight of weighed arithmetic mean = 11

3.If two or more quantities are added algebraically, the weight of the result is equal to the reciprocal of the sum of the reciprocals of the individual weights

θ = 3208’10” (weight 4) : ɸ = 2209’6” (weight 2)

Weight θ + ɸ = 54017’16” =

Weight θ - ɸ = 9059’4” =


Angle , P Weight

35020’10” 2

35020’12” 3

35020’8” 4

35020’10” 2

3

4

21

411

3

4

21

411

13


4.If a quantity of given weight is multiplied by a factor, the weight of the result is obtained by dividing its given weight by the square of the factor

θ = 42010’20” weight = 6 weight of 3 θ (= 126031’ )

5.If a quantity of given weight is divided by a factor, the weight of the result is obtained by multiplying its given weight by the square of the factor

θ = 42010’ 30” weight = 4 weight of θ (= 1403’30” ) = 4 x 32 = 36 36.If an equation is multiplied by its own weight, the weight of the resulting equation is equal

to the reciprocal of the weight of the equation A+ B = 98020’ 30” , weight 3/5 Weight of 3 (A+B) = 590 18” is equal to 1 or 5 5 (3/5) 3

7.The weight of an equation remains unchanged, if all the signs of the equation are changed or if the equation is added to or subtracted from a constant

A+ B = 80020’ , weight 3 Weight of 1800 – (A+B) or 99040’ is equal to 3


3

2

3

62

14


Rules of assigning weightage to field observation1. Weight of angle varies directly as the number of the observations made for the

measurement of that angle2. Weights vary inversely as the length of various routes in the case of lines of levels3. If an angle is measured a large number of times, its weight is inversely proportional to

the square of the probable error4. Corrections to be applied to various observed quantities are in inverse proportion to their

weights


15


DETERMINATION OF PROBABLE ERROR (PE)Case 1: Direct observation of equal weight on a single unknown quantity

Observation on a single quantity are made with equal weights, its most probable value is equal to the arithmetic mean

a. Probable error of single observation of unit weight

b. Probable error of single observation of weight w PE of single observation of unit weight = Es

√Weight √w

c. Probable error of the arithmetic mean,


16745.0

2

n

vEs

v- residual error n- number of observation

n

E

nn

VE s

m

)1(6745.0

2

16


Case 2: Direct observation of unequal weight on a single quantityThe most probable value of the observed quantity(N) is equal to the weighted arithmetic

mean of the observed quantitiesLet V1,V2,….,Vn are observed quantities with weight w1,w2,…,wn

a. Probable error of single observation of unit weight

b. Probable error of single observation of weight w PE of single observation of unit weight = Es =

√Weight √wc. . Probable error of weighted arithmetic mean,


n

nn

nn

nn

nn

www

VwVwVwN

imumVNwVNwVNw

VNvVNv

imumvwvwvw

...

....

min.....

........

min...

21

2211

2222

211

11

2222

211

16745.0

2

n

wvEs

)1(6745.0

2

nw

wv

)1(6745.0

2

nw

wv

17


Case 3: Probable error of computed quantities1.If a computed quantity is equal to sum or difference of the observed quantity plus or minus

a constant, the probable error of the computed quantity is the same as that of observed quantity

x = observed quantity; y = computed quantity ; k = constant ex – PE of the observed quantity : ey - corresponding PE of computed quantity

y = ± x ± k : ey = ex

2. If computed quantity is equal to an observed quantity multiplied by a constant the PE of computed quantity is equal to the pE of observed quantity multiplied by the constant

y = kx : ey = kex

3.If a computed quantity is equal to sum of two or more observed quantity, the PE of computed quantity is equal to the square root of sum of the square of PE of observed quantities

x1,x2,… be observed quantities : y – computed quantity

ex1,ex2,…..PE of observed quantity : ey - corresponding PE of computed quantity

Sivapriya Vijayasimhan.....

.....

22

21

21

xxy eee

xxy

18



4. If computed quantity is a function of an observed quantity, its probable error is obtained by multiplying the PE of the observed quantity with its differentiation with respect to that quantity

5.If a computed quantity is a function of two more observed quantity, its PE is equal to the square root of summation of the squares of PE of the observed quantity multiplied by its differentiation with respect to that of quantity

xy edx

dye

xfy

)(

2

2

2

1

,..21 ),(

dx

dye

dx

dyee

xxfy

xxy

19


Error Distribution to the Field MeasurementAccuracy is checked at the completion of work by computing closing errorClosing error is distributed to the observed quantity

1. Correction to be applied to an observation is inversely proportional to the weight of the observation

2. Correction to be applied to an observation is directly proportional to the square of the probable error

3. Correction to be applied to an observation is proportional to the length in case of line of levels


20


Normal EquationMost probable value of any unknown quantity is determinedFound by multiplying each equation by co-efficient of unknown whose normal equation is to

be found and by adding the equation thus formed

Round angles of equal weight, x + y + z = 360o = -d (say) The most probable value of ach angle can be obtained after applying error (‘e’) by applying

correction factor of 1/3 e to observed angle

One angle is measured directly and others indirectly, the error equation is e = (ax + by + cz + d)

For different values (x1,y1, z1) , (x2,y2,z2) etc, then

etc

By theory of least square should be minimum Sivapriya Vijayasimhan

dczbyaxe

dczbyaxe

2222

1111

22 )( dczbyaxe

21


Differentiate the equation, wr.t to x,y and z to obtain zero (Normal equation for x) ------------------------------------1 (Normal equation for y)-------------------------------------2 (Normal equation for z) ------------------------------------3The normal equation in x,y and z respectively are

If the observations are of equal weight, we derive the following rule for forming the normal equation

Rule I : To form a normal equation for each of the unknown quantities, multiply each observation equation by the algebraic co-efficient of that unknown quantity in that equation and add the result

(weight w1)

(weight w2)

(weight wn)


0)(

0)(

0)(

dczbyaxc

dczbyaxb

dczbyaxa

....)(

....)(

....)(

222111

222111

222111

dczbyaxdczbyaxcdczbyaxc

dczbyaxdczbyaxbdczbyaxb

dczbyaxdczbyaxadczbyaxa

dczbyaxe

dczbyaxe

dczbyaxe

nnnn

'2222

1111

22


By theory of least squares

Differentiating the above equation (Normal equation to x) (Normal equation to y) (Normal equation to z)The normal equation in x,y and z respectively are

Rule II : To form a normal equation for each of the unknown quantities, multiply each observation equation by the product of the algebraic co-efficient of that unknown quantity in that equation and weight of that observation and add the result


22 )(we dczbyaxw

0)(

0)(

0)(

dczbyaxwc

dczbyaxwb

dczbyaxwa

....)(

....)(

....)(

22221111

22221111

22221111

dczbyaxwdczbyaxwcdczbyaxwc

dczbyaxwdczbyaxwbdczbyaxwb

dczbyaxwdczbyaxwadczbyaxwa

23


Determination of the most probable values1.Direct observation of equal weights2.Direct observation of unequal weights3.Indirect observed quantities involving unknowns of equal weights4.Indirect observed quantities involving unknowns of unequal weights5.Observation equations accompanied by condition equation

Case I :Direct observation of equal weightsMost probable value of the directly observed quantity of equal weights is equal to the

arithmetic mean of the observed values

Case 2:Direct observation of unequal weightsMost probable value of an observed quantity of unequal weights is equal to the weighed

arithmetic mean of the observed quantities


n

nn

www

VwVwVwM

...

...

21

2211

n

VVVM n

...21

24


Case 3 and 4 : Indirect observed quantities unknowns of equal weights or unequal weightsWhen the unknowns are independent of each other, their most probable value can be found

by forming normal equations and solving the unknowns

Case 5: Observation equations accompanied by condition equationObservation equations are accompanied by one or more condition equations, the latter may

be reduced to an observation equationNormal equation is formed by remaining unknowns


25


Methods of correlatesThese are unknown multiples or independent constants used for finding most probable values

of unknownsA,B,C and D are angles measured at station closing the horizon. w1, w2, w3 and w4 are the weights respectively

E – total residual error

A + B + C + D – 360o = E e1, e2, e3 and e4 are the corrections to be applied

Differentiate the above two equations,

Multiply equations by –λ1 and add the result to equation 4


imumewewewewwe

Eeeeee

min244

233

222

211

2

4321

0

0

44433322211

4321

eeweeweeweewewe

eeeee

--------------------------------------------------------------- 1 ------------------------------ 2

---------------------------------------------- 3 - --------------------- 4

0)()()()(

0

1444133312221111

444333222111

41312111

eweeweeweewe

eeweeweeweew

eeee

26



4

14

3

13

2

12

1

11

443322111

we

we

we

we

ewewewew

Ewwww

Ewwww

43211

4

1

3

1

2

1

1

1

1111

δe1,δe2,δe3 andδe4 are definite and independent , their co-efficient vanishes

To find λ1 , substitute above values in equation 1

27


Triangulation and AdjustmentsConditions imposed by station of observation – Station adjustmentConditions imposed by figure – Figure adjustment1. Single angle adjustment2. Station adjustment3. Figure adjustment

Single angle adjustmentCorrections applied are inversely proportional to weight and directly proportional to square

of probable errorsMeasurement of angle with equal weights : most probable values is equal to arithmetic mean

of observationWeighted observations: most probable value of the angle is equal to weighted arithmetic

mean of observed angle

Station adjustment1.Horizon is closed with angles of equal weights2.Horizon is closed with angles from unequal weights3.Several angles are measured at station individually and in combination


28


Horizon is closed with angles of equal weights A,B and C are measured horizon angles A+B+C = 360o

If this condition is not satisfied, error is distributed equally

Horizon is closed with angles from unequal weightsError is distributed among the angles inversely as the respective weights

Several angles are measured at station individually and in combinationForm normal equation for unknowns and solve simultaneously


29


Figure AdjustmentDetermination of the most probable value of the angles involved in any geometrical figure so

as to fulfil geometric condition

Triangulation system of following geometrical figures:1.Triangules2.Qudrilaterals3.Polygons with central figure

Figure adjustment of a triangle Simple figure having three interior angles, measured independently and sum is equal to 180o

If not 180o , angles are distributedCorrected angles is used to calculate other two sides of triangle if length of one side is known.


30


Adjustment of chain of triangle 1.Station Adjustment2.Figure Adjustment

Station Adjustment

If discrepancy is note, it is distributed equally to component angles

Figure AdjustmentEach triangle is taken separately for figure adjustmentIn ABC,In ACD,In CDE,Angles of equal weights , discrepancy is distributed equally to 3 angles else, it is distributed in inverse proportion to their

weights


0

0

0

0

0

3601413

360121110

3609876

36054

360321

0180641 01801072 018013118

31


Adjustment of two connected of triangleTotal 8 angles and 4 independent condition equation

C1,C2,D1 and D2 are independent unknowns

Remaining are dependants Using normal equation, 4 unknowns A,B,C and D can be expressed in terms if independent.


DDD

CCC

DCB

DCA

21

21

022

011

180

180

21

21

220

110

)(180

)(180

DDD

CCC

DCB

DCA

32


Triangle adjustment with central station ABC, consider a central station OAngles measured are θ1, θ2 .....θ9

θ7, θ8 and θ9 are central angles

θ1, θ2 and θ3 are left angles : θ4, θ5 and θ6 are right angles

c1,c2... be the corrections to angles θ1, θ2 etc

f1,f2.. be the tabular difference for 1” for log sin θ1, log sin θ2 etc

Equation of condition1. Apex condition : Sum of angles around the central station or common vertex must be

equal to 3600 2. Triangle Condition : Sum of angles of each triangle must be equal to 1800

3. Conditional equation generated by lines AO,BO and CO should satisfy by the angles along the periphery of triangle

Sivapriya Vijayasimhan4

3

6

5

1

2

sin

sin,

sin

sin,

sin

sin,

BOCOBCO

COAOACO

BOAOABO

------------- 1

------------- 2

-------------- 3

33



6.42531

6

5

4

3

1

2

6

5

4

3

sinsin.sinsin.sin.sin

sin

sin

sin

sin

sin

sin

sin

sin

sin

sin

BOBO

BOAO (Sub 3 in 1) ----------------------------4

( 4 =1)

This equation is called side equation sum of log sine of left angles = sum of log sine of right angles (log sine condition)

Apex condition : Triangle condition :

Log sine condition:

(M –units of 7th decimal place of log)By least square condition

RL sinlogsinlog

1987 Kccc

4965

3843

2721

Kccc

Kccc

Kccc

Mcfcfcfcfcfcf 665544332211

min29

28

27

26

25

24

23

22

21 ccccccccc

34


Geodetic Quadrilateral AdjustmentIn geodetic quadrilateral all the interior angles are measured independentlyIf the quadrilateral size is small, it may be considered as a plane quadrilateralIf the quadrilateral size is large, the spherical excess has to be calculated separatelyA correction of 1/3 spherical excess is applied to each angle of triangles

Three methods,1. Rigorous method of least square (angle and side equation)2. Approximate method3. Method of equal shifts


35


Methods of Equal ShiftsClosed polygon of five sides with central station OFigure adjustment done adopting method of equal shifts

- Any shift which is necessary to satisfy the local equation should be same for each triangle of polygon- Any shift necessary to satisfy the side equation should be same for each triangle

Equation of condition1.Figure Equation Sum of triangles = 180 o

2.Station or local equation Sum of angles at a station = 360 o

3.Side equation


RL sinlogsinlog

36


Level NetsInterconnecting network of level circuits formed by level lines inter-connecting 3 or more BM

Most probable values(MPV) of several differences of elevations among the BM may be obtained using, 1. method of correlates and 2. method of normal equation

MPV of BM :Direct observed elevations by method of normal equation The weights to be assigned to the observed difference elevation of the ends of a

connected line is taken as inversely proportional to length of the line


37


Figure AdjustmentTo determine the most probable value of the angles involved in any geometrical figure so as to

fulfil the geometric requirementGeometrical figures adopted in triangulation system are, 1. Triangles 2. Quadrilateral 3. Polygons with central station

Triangle Adjustment -Simple figure formed by connecting three points by straight lines or by arcs of great circle.The figure formed by joining three points by straight line is called plane triangleThe figure formed by lines connecting any three points on the mean of the earth is called

spherical triangle

Plane triangle Spherical triangle


38

UNIT III SURVEY ADJUSTMENTS Sum of three angles ( ) = 1800 in plane triangle = 1800 + spherical excess Let A,B and C be the observed angles c1, c2 and c3 be the corresponding corrections c be the total correction or error of closure

w1, w2 and w3 be the relative weights of A,B and C

E1, E2 and E3 be probable error of A,B and C

n1, n2 and n3 be number of observations for angles A,B and C respectively

1.Equal Weight CorrectionRule : If all angles of a triangle are of equal weight, the discrepancy is distributed to all the three angles

2.Inverse Weight CorrectionRule: If all angles of a triangle are of unequal weight, the discrepancy is distributed to all the angles in

inverse proportion to the weights


andBA ^^ , ^CandBA oo^^ , ^

0C

cccc3

1321

c

www

wc

wwwccc

321

11

321321

111

1

1:

1:

1::

c

www

wc

c

www

wc

321

33

321

22

111

1

111

1

39


3.Inverse CorrectionRule : If the weights of observations are not given, the discrepancy is distributed to all the

three angles in inverse proportion to their number of observations


c

nnn

nc

c

nnn

nc

c

nnn

nc

nnnccc

321

33

321

22

321

11

321321

111

1

111

1

111

1

1:

1:

1::

40


4.Inverse Square CorrectionRule : The discrepancy is distributed to all angles in inverse portion to the square of the square

of the number of observations


c

nnn

nc

c

nnn

nc

c

nnn

nc

nnnccc

2

3

2

2

2

1

2

33

2

3

2

2

2

1

2

22

2

3

2

2

2

1

2

11

2

3

2

2

2

1321

111

1

111

1

111

1

1:

1:

1::

41


5.Probable Error Square CorrectionRule : If the probable errors of each angle of a triangle are known, then the discrepancy is

distributed to all angles in direct proportions to the square of probable error

6.Gauss’s RuleRule : This rule is applied when the weights of the observations are not directly known. If the

residuals error of each observation is known the weights can be calculated by the Gauss’s rule given by the following expression:


cEEE

Ec

cEEE

Ec

cEEE

Ec

EEEccc

23

22

21

23

3

23

22

21

22

2

23

22

21

21

1

23

22

21321 ::::

2

2 2

v

nw

w-weight to be assigned to a quantity

n – is the number of observation made for the quantity - sum of squares of residuals2v

42



)(2

)(2

)(2

323

23

3

222

22

2

121

21

1

sayKv

nw

sayKv

nw

sayKv

nw

With the knowledge of weights, the corrections are applied by rule 2

cKKK

KC

cKKK

KC

cKKK

KC

KKKwww

ccc

321

33

321

22

321

11

321321

321 ::1:

1:

1::

43


7.Spherical TriangleSum of three angles of a spherical triangle always exceeds 180 deg (spherical excess)

i.Spherical Excess(Es)- Depends upon area of triangle- It is ignored when the sides of triangles is less than 3.5 km (approximately to 1” for every

196.75 sq.km)Exact value of spherical excess degree (Es

0)

- area of spherical triangle in sq.m or sq.kmR- radius of the sphere of earth in m or km


"1sin

180

2"

2

0

RE

RE

s

os

ondsEs sec197

"

44


ii.Area of Spherical triangleTo compute area of spherical triangle, it is assumed as plane

a- known sideA,B and C - observed anglesiii. Side of a Spherical triangleUsing spherical trigonometry

A0,B0 and C0 – adjusted angles of spherical triangle

a= B’C’ : b = A’C’ : c = A’B’a1- angle subtended by side B’C’ at the centre of sphere

b1- angle subtended by side A’C’ at the centre of sphere

c1- angle subtended by side A’B’ at the centre of sphere


A

CBa

sin

sinsin

2

1 2

45


Step 1:Central angle a1 of B’C’ (= a) arc = R x central angle

a1 in degree

R radius of earth

Step 2:Knowing a1 , the central angles b1 and c1 by sine rule

Step 3:Knowing the central anglesb1 and c1 the corresponding length of arc C’A’ (=b) and A’B’(=c) are

calculated


Ra

R

arcleCentralang

0

1

180

0

011

0

011

sin

sinsinsin

sin

sinsinsin

A

Cac

A

Bab

01

01

180

180Rc

c

Rbb

46


8. Plane Triangle The sum of angles of triangle seldom happens to be equal to 180 degAfter correction of angles., sides of triangle may be computed from a known side and three

angles using sine ruleOne measurement is known by direct measurement as a base line or known preceding

computations


SURVEY ADJUSTMENTS. UNIT III SURVEY ADJUSTMENTS CONTENTS Errors Sources, precautions and corrections...

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Transcript of SURVEY ADJUSTMENTS. UNIT III SURVEY ADJUSTMENTS CONTENTS Errors Sources, precautions and corrections...