Surveying II

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Surveying II
Muhammed SAHIN Ergin TARI
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CONTENTS

• Basic definitions
• Relations between deviations
• Error propagation law
• Weight
• Weight propagation law
• Numerical examples

Compiled by Himmet KARAMAN
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Precision

• Degree of consistency between measurements and
  is based on the size of the discrepancies in a
  data set. The degree of precision attainable
  (achievable) is dependent on the stability of the
  environment during the time of measurement, the
  quality of the equipment used to make the
  measurements, and the observers skill with the
  equipment and measurement procedures.

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Accuracy

• Measure of the absolute nearness of a measured
  quantity to its true value. Since the true value
  of a quantity can never be determined, accuracy
  is always an unknown.

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Precision Versus Accuracy
In general, when making measurements, data such
as those shown in Figure (b) and (d) are
undesirable. Rather, results similar to those
shown in Figure (a) are preferred. However, in
making measurements the results of Figure (c) can
be just as acceptable if proper steps are taken
to correct for the presence of the systematic
errors. This correction would be equivalent to
the marksman realigning the sights after taking
the shots.
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Basic Definitions

• True Value
• a quantitys theoratically correct or exact
  value.
• ATTENTION True value can never be determined
• Error (e)
• the difference between any individual measured
  quantity and its true value. The true value is
  simply the populations arithmetic mean if all
  repeated measurements have equal precision.
  Errors are expressed as ei yi µ
• where yi is the individual measurement associated
  with error ei and µ is the true value for that
  quantity.

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Basic Definitions

• Most Probable Value ( y ) is derived from a
  sample set of data rather than the population,
  and simply the mean if the repeated measurements
  have the same precision.
• Residual (?)is the difference between any
  individual measured quantity and the most
  probable value for that quantity. The
  mathematical expression for a residual is ?i
  y yi
• Degrees of Freedom is the number of observations
  that are in excess of the number necessary to
  solve for the unknowns. In other words, the
  number of degrees of freedom equals the number of
  redundant (unnecessary) observations.

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Basic Definitions

• Variance (s2) is a value by which the precision
  for a set of data is given. Population variance
  applies to a data set consisting of an entire
  population. It is the mean of the squares of the
  errors and given by
• S ei2
• s2
• Sample variance applies to a sample set of data.
  It is unbiased estimate for the population
  variance given above, and is calculated as
  
• S ?i2
• S2

n
i1
n
n
i1
n - 1
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Basic Definitions

• Standard Error (s) is the square root of the
  population variance.
• Where n is the number of measurements and Sni1
  ei2 is the sum of the squares of the errors.

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Basic Definitions

• Standard Deviation (S) is the square root of the
  sample variance.
• Where S is the standard deviation, n 1 is the
  degrees of freedom or number of redundancies, and
  Sni1 ?i2 is the sum of the squares of the
  residuals.

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Basic Definitions

• Standard Deviation of the Mean Because all
  measured values contain errors, the mean that is
  computed from a sample set of measured values
  will also contain errors.
• Sy S

vn
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The Law of Error Propagation

• Often the standard deviation of a function of
  measured quantities is required in addition to
  the standard deviation of a single observation.
  This can be obtained by applying the law of error
  propagation.

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The Law of Error Propagation

• Linear Functions
• The standard deviation sx of the function x
  l1 l2
• is to be found, using the given observations l1
  and l2 with their standard deviations s1 and s2.
  In order to compute sx, one has to utilize the
  definition given in equation of standard error
  with the assumption that the values li in
  equation of (x l1 l2) are the means of ?
  original observations with true errors e1,
  e1,..., e1(?) or e2, e2,..., e2(?) where ?
  is infinitely large. This represents ? equations
  of the form ex e1 e2.

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The Law of Error Propagation

• By squaring, adding, and dividing by ?, one
  obtains
• Sex2 Se12 Se22 Se1e2
• ? ? ? ?
• The first three expression represent sx2, s12,
  s22 according to standard error equation. The
  last term is together with the division by ?,
  which is very large, indicates that this term
  approaches zero. Therefore, we can write sx2
  s12 s22.

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Example

• -Three adjanted distances along the same line
  were measured independently with the following
  results x151,00m with s10,05m x236,50m with
  s20,04m x326,75m with s30,03m. Compute the
  total distance and its standard deviation.

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Example

• - y x1 x2 x3 114,25m

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Example

• -If in example before, all three distances were
  measured with a standard deviation of sx0,05m,
  what would sy be?
• -

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The Law of Error Propagation

• If one replaces s by its estimator s, similar to
  the transition from standard error equations and
  standard deviation equations the following rule
  is obtained for estimating the standard deviation
  of the sum in equation of (xl1l2).
• sx2 s12 s22.

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The Law of Error Propagation

• Applying the same principle to the following
  function
• x a1 l1 a2 l2 ... an ln (1)
• leads to
• sx2 a12 s12 a22 s22 ... an2 sn2 (2)
• The following special cases should be noted
• For s1 s2 ... sn s sn2 s2
  Sa2
• If in addition all ai are either 1 or -1, then
• sx2 ns2 or sx svn

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The Law of Error Propagation

• This case occurs when levelling or measuring
  distances with tapes. The rule is expressed as
• If several single observations of equal accuracy
  are combined to a sum or difference, then the
  standard deviation of the result increases with
  the square root of the number of single
  observations.

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The Law of Error Propagation

• c) If arithmetic mean equation is written as
• x l1 l2 ... ln
• n n n
• and the all si equal s, then the following value
  of sx is obtained according to equation (2)
• sx2 s2 s2 ... s2 or sx s
• n2 n2 n2
• which reaffirms the value given in the standard
  deviation equation. This rule is expressed as

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The Law of Error Propagation

• If the same object is measured n times with the
  same accuracy, then the standard deviation of the
  arithmetic mean is reduced by the square root of
  the number of the repetition.

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The Law of Error Propagation

• Nonlinear Functions
• These functions are usually linearized with the
  aid of Taylor Series Expansion. Then equation (2)
  can again be applied. The law of error
  propagation in its most general form is
  therefore
• For x f(l1, l2, ...,ln)
• sx2 ( )2s12 ( )2s22... ( )2sn2.

?f ?l1
?f ?l1
?f ?l1
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Example

• -In trigonometric leveling, the slope distance is
  s 50,00m with ss0,05m and ß30o00with
  s?0030. Compute h and sh while assuming s and
  ß to be uncorrelated.
• - h s.sin ß (50,00).(0,5) 25,00m

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Example

• - The area of a rectangular parcel of land is
  required together with its standard deviation.
  The length is a100m with sa0,10m, and the width
  is b40m with sb0,08m. (Assume the mutual
  variation between the a and b is 0 or they are
  uncorrelated)

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Example

• -

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Weights

• If a measured quantity has been observed several
  times with different accuracies, these accuracy
  relations (weights) have to be considered when
  computing the mean. An observation is given the
  weight p if it has the same standard deviation as
  the arithmetic mean of p real or fictitious (not
  real, imaginary) standard observations with unit
  weight.

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Weights

• For the measurements L1, L2, ...,Ln with weights
  p1, p2, ..., pn follows from equation of the
  standard deviation of the mean
• p1p2 ... pn
• This means that the weights are inversely
  proportional to the squares of the standard
  deviations.

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Weights

• The standard deviation of a measurement with
  weight 1 is called the standard error of unit
  weight and is usually denoted as s0.
• In order to compute the weight pi of an
  observation Li with a standard deviation si the
  following equation is used

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Weights

• The unit weight is chosen such that the
  individual weights deviate as little as possible
  from 1. The following weights are often used for
  unity
• Weight of a distance of 100 m, of a levelling
  loop of 1 km, or of an angle observed in direct
  and reverse.

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The Law of Propagation of Weights

• With the values derived from in the
  equation of weight the law of propagation of
  weights follows directly from the law of error
  propagation
• If an observation Li is multiplied by the square
  root of its weight, then, according to equation
  above has weight 1.
• This expression is referred to as the
  standardized or normed variable.

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Example

• - A distance is measured independently by two
  observers to be x1110,00m and x2110,80m. If the
  weights of these two measurements are p12 and
  p23, respectively, and its best estimate is the
  weighted mean given by
• compute the standard deviation of .

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Example

• - Taking the weights as reciprocals of the
  variances,

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Example

• (contd)- which means that the weight of the
  weighted mean of two observations is equal to the
  sum of their weights,
• or
• Finally