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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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
4
UNIT III SURVEY ADJUSTMENTS
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.
Sivapriya Vijayasimhan
5
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
0180 CBA
6
UNIT III SURVEY ADJUSTMENTS
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.
Sivapriya Vijayasimhan
7
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
Size of error
1
Σ6745.0
2
n
vEs
Difference in any single observation and mean of series
No. Of observation in series
8
UNIT III SURVEY ADJUSTMENTS
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
'
'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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
Angle , P Weight
35020’10” 1
35020’12” 1
35020’8” 1
35020’10” 1
12
UNIT III SURVEY ADJUSTMENTS
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” =
Sivapriya Vijayasimhan
Angle , P Weight
35020’10” 2
35020’12” 3
35020’8” 4
35020’10” 2
3
4
21
411
3
4
21
411
13
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
3
2
3
62
14
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
15
UNIT III SURVEY ADJUSTMENTS
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,
Sivapriya Vijayasimhan
16745.0
2
n
vEs
v- residual error n- number of observation
n
E
nn
VE s
m
)1(6745.0
2
16
UNIT III SURVEY ADJUSTMENTS
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,
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
UNIT III SURVEY ADJUSTMENTS
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
20
UNIT III SURVEY ADJUSTMENTS
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
UNIT III SURVEY ADJUSTMENTS
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)
Sivapriya Vijayasimhan
0)(
0)(
0)(
dczbyaxc
dczbyaxb
dczbyaxa
....)(
....)(
....)(
222111
222111
222111
dczbyaxdczbyaxcdczbyaxc
dczbyaxdczbyaxbdczbyaxb
dczbyaxdczbyaxadczbyaxa
dczbyaxe
dczbyaxe
dczbyaxe
nnnn
'2222
1111
22
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
22 )(we dczbyaxw
0)(
0)(
0)(
dczbyaxwc
dczbyaxwb
dczbyaxwa
....)(
....)(
....)(
22221111
22221111
22221111
dczbyaxwdczbyaxwcdczbyaxwc
dczbyaxwdczbyaxwbdczbyaxwb
dczbyaxwdczbyaxwadczbyaxwa
23
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
n
nn
www
VwVwVwM
...
...
21
2211
n
VVVM n
...21
24
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
25
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
28
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
29
UNIT III SURVEY ADJUSTMENTS
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.
Sivapriya Vijayasimhan
30
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
0
0
0
0
0
3601413
360121110
3609876
36054
360321
0180641 01801072 018013118
31
UNIT III SURVEY ADJUSTMENTS
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.
Sivapriya Vijayasimhan
DDD
CCC
DCB
DCA
21
21
022
011
180
180
21
21
220
110
)(180
)(180
DDD
CCC
DCB
DCA
32
UNIT III SURVEY ADJUSTMENTS
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
UNIT III SURVEY ADJUSTMENTS
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
35
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
RL sinlogsinlog
36
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
37
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
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
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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:
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
Sivapriya Vijayasimhan
)(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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
"1sin
180
2"
2
0
RE
RE
s
os
ondsEs sec197
"
44
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
A
CBa
sin
sinsin
2
1 2
45
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan
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
UNIT III SURVEY ADJUSTMENTS
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
Sivapriya Vijayasimhan