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SURVEY ADJUSTMENTS

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**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

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**Errors Sources of Errors Measurement 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 Errors Two 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

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**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

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**Quantities Observations**

Independent Quantity : Independent of values of other quantities .Example: Reduced level Conditioned Quantity : Dependent upon the values of one or more quantity. Also called as dependent quantity Observed value of quantity: Value obtained after all correction of errors True Value of Quantity: Absolutely free from all errors .It is indeterminate and it is never known Observations Direct Observation: Measured directly upon desired quantity Eg. Measurement of single angle Indirect Observation: Observed value is deduced from measurement of some related quantities. Eg. Measurement of a summation angle for the sum of an angle by repetition Weight of an Observation : Number giving an indication of its precision and trustworthiness when making comparison between several quantities of different worth Weightage = 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

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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 weights Most Probable Error: Quantity which is added to and subtracted from the most probable value which fixes the limit True Error: Difference between the true value of an quantity and its observed value Residual Error :Difference between most probable value of a quantity and its observed value Equation Observation Equation: Relation between observed quantity and its numerical value Conditioned 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

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**Law of Accidental Errors**

Law of probability Errors in form of equation which is used to compute the probabale values or precision of a quantity Features Small errors tend to be more frequent than large one +ve and –ve of same size happen with equal frequency Large errors occur impossible Probable error of single measurement Size of error Difference in any single observation and mean of series No. Of observation in series Sivapriya Vijayasimhan

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**PRINCIPLES OF LEAST SQUARE**

Probable error of an average, Probable error of sum, Where, E12 , E22, E32, En2 are probable errors PRINCIPLES OF LEAST SQUARE In 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 values X- most probable value E1,E2…..En be the respective error of the observed values M – arithmetic value N- number of observation Sivapriya Vijayasimhan

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

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) (ΣR/n = 0) Sivapriya Vijayasimhan

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**------------------------2**

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 Substitute nM=ΣV Substitute M=ΣV/n Sivapriya Vijayasimhan

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

LAWS OF WEIGHTS 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” 4 Weight of arithmetic mean = Number of observations = 4 Angle , P Weight 35020’10” 1 35020’12” 35020’8” Sivapriya Vijayasimhan

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2.Weight of the weighted arithmetic mean is equal to the sum of the individual weights Sum of individual weights = =11 Weighted arithmetic mean = 35020’ + 1 (10”+ 12”+ 8”+ 10”) = 35020’3.64” 11 Weight 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 Sivapriya Vijayasimhan

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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 (= ’ ) 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 = 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) = ” 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

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**Rules of assigning weightage to field observation**

Weight of angle varies directly as the number of the observations made for the measurement of that angle Weights vary inversely as the length of various routes in the case of lines of levels If an angle is measured a large number of times, its weight is inversely proportional to the square of the probable error Corrections to be applied to various observed quantities are in inverse proportion to their weights Sivapriya Vijayasimhan

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**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 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, v- residual error n- number of observation Sivapriya Vijayasimhan

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**Case 2: Direct observation of unequal weight on a single quantity **

The most probable value of the observed quantity(N) is equal to the weighted arithmetic mean of the observed quantities Let V1,V2,….,Vn are observed quantities with weight w1,w2,…,wn Probable error of single observation of unit weight Probable error of single observation of weight w PE of single observation of unit weight = Es = √Weight √w c. . Probable error of weighted arithmetic mean, Sivapriya Vijayasimhan

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**Case 3: Probable error of computed quantities**

1.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

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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 Sivapriya Vijayasimhan

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**Error Distribution to the Field Measurement **

Accuracy is checked at the completion of work by computing closing error Closing error is distributed to the observed quantity Correction to be applied to an observation is inversely proportional to the weight of the observation Correction to be applied to an observation is directly proportional to the square of the probable error Correction to be applied to an observation is proportional to the length in case of line of levels Sivapriya Vijayasimhan

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Normal Equation Most probable value of any unknown quantity is determined Found 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

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**Differentiate the equation, wr**

Differentiate the equation, wr.t to x,y and z to obtain zero (Normal equation for x) (Normal equation for y) (Normal equation for z) The 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

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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

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**Determination of the most probable values**

1.Direct observation of equal weights 2.Direct observation of unequal weights 3.Indirect observed quantities involving unknowns of equal weights 4.Indirect observed quantities involving unknowns of unequal weights 5.Observation equations accompanied by condition equation Case I :Direct observation of equal weights Most 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 weights Most probable value of an observed quantity of unequal weights is equal to the weighed arithmetic mean of the observed quantities Sivapriya Vijayasimhan

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Case 3 and 4 : Indirect observed quantities unknowns of equal weights or unequal weights When 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 equation Observation equations are accompanied by one or more condition equations, the latter may be reduced to an observation equation Normal equation is formed by remaining unknowns Sivapriya Vijayasimhan

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Methods of correlates These are unknown multiples or independent constants used for finding most probable values of unknowns A,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

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**To find λ1 , substitute above values in equation 1**

δe1,δe2,δe3 andδe4 are definite and independent , their co-efficient vanishes To find λ1 , substitute above values in equation 1 Sivapriya Vijayasimhan

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**Triangulation and Adjustments**

Conditions imposed by station of observation – Station adjustment Conditions imposed by figure – Figure adjustment Single angle adjustment Station adjustment Figure adjustment Corrections applied are inversely proportional to weight and directly proportional to square of probable errors Measurement of angle with equal weights : most probable values is equal to arithmetic mean of observation Weighted observations: most probable value of the angle is equal to weighted arithmetic mean of observed angle 1.Horizon is closed with angles of equal weights 2.Horizon is closed with angles from unequal weights 3.Several angles are measured at station individually and in combination Sivapriya Vijayasimhan

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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 weights Error is distributed among the angles inversely as the respective weights Several angles are measured at station individually and in combination Form normal equation for unknowns and solve simultaneously Sivapriya Vijayasimhan

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Figure Adjustment Determination 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.Triangules 2.Qudrilaterals 3.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 distributed Corrected angles is used to calculate other two sides of triangle if length of one side is known. Sivapriya Vijayasimhan

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**Adjustment of chain of triangle**

1.Station Adjustment 2.Figure Adjustment Station Adjustment If discrepancy is note, it is distributed equally to component angles Figure Adjustment Each triangle is taken separately for figure adjustment In 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

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**Adjustment of two connected of triangle**

Total 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

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**Triangle adjustment with central station**

ABC, consider a central station O Angles measured are θ1, θ 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 condition Apex condition : Sum of angles around the central station or common vertex must be equal to 3600 Triangle Condition : Sum of angles of each triangle must be equal to 1800 Conditional equation generated by lines AO,BO and CO should satisfy by the angles along the periphery of triangle Sivapriya Vijayasimhan

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**This equation is called side equation **

(Sub 3 in 1) ( 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 Sivapriya Vijayasimhan

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**Geodetic Quadrilateral Adjustment**

In geodetic quadrilateral all the interior angles are measured independently If the quadrilateral size is small, it may be considered as a plane quadrilateral If the quadrilateral size is large, the spherical excess has to be calculated separately A correction of 1/3 spherical excess is applied to each angle of triangles Three methods, Rigorous method of least square (angle and side equation) Approximate method Method of equal shifts Sivapriya Vijayasimhan

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**Methods of Equal Shifts**

Closed polygon of five sides with central station O Figure 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 condition 1.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

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Level Nets Interconnecting 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

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Figure Adjustment To determine the most probable value of the angles involved in any geometrical figure so as to fulfil the geometric requirement Geometrical 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 triangle The figure formed by lines connecting any three points on the mean of the earth is called spherical triangle Plane triangle Spherical triangle Sivapriya Vijayasimhan

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Sum of three angles ( ) = 1800 in plane triangle = 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 Correction Rule : If all angles of a triangle are of equal weight, the discrepancy is distributed to all the three angles 2.Inverse Weight Correction Rule: 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

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3.Inverse Correction Rule : 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

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**4.Inverse Square Correction **

Rule : The discrepancy is distributed to all angles in inverse portion to the square of the square of the number of observations Sivapriya Vijayasimhan

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**w-weight to be assigned to a quantity **

5.Probable Error Square Correction Rule : 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 Rule Rule : 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: w-weight to be assigned to a quantity n – is the number of observation made for the quantity - sum of squares of residuals Sivapriya Vijayasimhan

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

Sivapriya Vijayasimhan

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**i.Spherical Excess(Es)**

7.Spherical Triangle Sum 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 sq.km) Exact value of spherical excess degree (Es0) - area of spherical triangle in sq.m or sq.km R- radius of the sphere of earth in m or km Sivapriya Vijayasimhan

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**ii.Area of Spherical triangle**

To compute area of spherical triangle, it is assumed as plane a- known side A,B and C - observed angles iii. Side of a Spherical triangle Using 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

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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

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8. Plane Triangle The sum of angles of triangle seldom happens to be equal to 180 deg After correction of angles., sides of triangle may be computed from a known side and three angles using sine rule One measurement is known by direct measurement as a base line or known preceding computations Sivapriya Vijayasimhan

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