How Errors Propagate
• Error in a Series
• Errors in a Sum
• Error in Redundant Measurement
Error in a Series
• Describes the error of multiple measurements with identical
standard deviations, such as measuring a 1000’ line with using
a 100’ chain.
•Eseries  E n
• Esum is the square root of the sum of each of the individual
measurements squared
• It is used when there are several measurements with differing
standard errors
Esum   E1  E2  E3  ...  En
2
2
2
2
Error in Redundant Measurements
• If a measurement is repeated multiple times, the accuracy
increases, even if the measurements have the same value
Ered.meas.  
E
n
If you learn one thing…
• With Errors of a Sum (or Series), each additional variable
increases the total error of the network
• With Errors of Redundant Measurement, each redundant
measurement decreases the error of the network.
• As the network becomes more complicated, accuracy can be
maintained by increasing the number of redundant
measurements
Introduction to Adjustments
• Adjustment - “A process designed to remove inconsistencies in
measured or computed quantities by applying derived
corrections to compensate for random, or accidental errors,
such errors not being subject to systematic corrections”.
Introduction to Adjustments
• Common Adjustment methods:
 Compass Rule
 Transit Rule
 Crandall's Rule
 Rotation and Scale (Grant Line Adjustment)
 Least Squares Adjustment
Weighted Adjustments
• Weight - “The relative reliability (or worth) of a quantity as
compared with other values of the same quantity.”
Weighted Adjustments
• The concept of weighting measurements to account for
different error sources, etc. is fundamental to a least
squares adjustment.
• Weighting can be based on error sources, if the error of
each measurement is different, or the quantity of readings
that make up a reading, if the error sources are equal.
Weighted Adjustments
• Formulas:
•W  (1  E2) (Error Sources)
• C  (1  W) (Correction)
•W  n (repeated measurements of the same value)
• W  (1  n) (a series of
• measurements)
Weighted Adjustments
• Example
A = 4324’36”, 2x
A
B = 4712’34”, 4x
C = 8922’20”, 8x
Perform a weighted
adjustment based on the
above data
C
B
Example
Mean
Value
Rel. Corr.
Corrections
Adjusted
Value
A
2
43 24’ 36”
4/
7
4/
7
X 30” = 17”
43 24’ 53”
B
4
47 12’ 34”
2/
7
2/
7
X 30” = 09”
47 12’ 43”
1/
7
X 30” = 04”
89 22’ 24”
= 30”
180 00’ 00”
The relative correction for the three angles are 1 : 2 : 4, the inverse proportion to
the number of turned angles. This is the first set of relative corrections.
The sum of the relative corrections is 1 + 2 + 4 = 7 , This is used as the
denominator for the second set of corrections. The sum of the second set of
relative corrections shall always equal 1. The second set is used for corrections.
C
8
89 22’ 20”
1/
7
TOTALS
17959’ 30”
7/
7
Introduction to Least Squares Adjustment
• A rigorous statistical adjustment of survey data based on
the laws of probability and statistics
• Provides simultaneous adjustment of all measurements
• Measurements can be individually weighted to account for
different error sources and values
• Minimal adjustment of field measurements
Least Squares Adjustment
• A Least Squares adjustment distributes random errors
according to the principle that the Most Probable Solution is
the one that minimizes the sums of the squares of the
residuals.
• This method works to keep the amount of adjustment to
the observations and, ultimately the ‘movement’ of the
coordinates to a minimum.
Least Squares
• The Iterative Process
•
•
•
•
•
Creates a calculated observation for each field observation by
inversing between approximate coordinates.
Calculates a "best fit" solution of observations and compares them
to field observations to compute residuals.
Updates approximate coordinate values.
Calculates the amount of movement between the coordinate
positions prior to iteration and after iteration.
Repeats steps 1 - 4 until coordinate movement is no greater than
selected threshold.
Least Squares
• Four component that need to be addressed prior to
performing least squares adjustment
1.
2.
3.
4.
Errors
Coordinates
Observations
Weights
Each Observation Requires an Associated Weight
• Weight = Influence of the Observation on Final Solution
• Larger Weight - Larger Influence
• Weight = 1/σ2
• σ = Standard Deviation of the Observation
• The Smaller the Standard Deviation the Greater the Weight
Weighting Methods
• Observational Group
• Least Desired
• Angles weighted at accuracy of total station
• Individually weighted
• Best
• Std dev. Of field observation used as weight
Least Square Example
Perpendicular offsets:
1 = (0,0)
2 = (100,100)
3 = (200, 400)
This example - Perpendicular offset = 141.421’
1: r = 0, r sq. = 0
2: r = 0, r sq. = 0
3: r = 141.421, r sq. = 20,000
Sum r sq. = 20,000
1: r = 63.246, r2 = 4,000
2: r = 0, r2 = 0
3: r = 0, r2 = 0
Sum r2 = 4000
1, 2 & 3: r = 22, r2 = 484
Sum r2 = 3*484 = 1452
This has the lowest Sum r2 therefore is best result so far
Actual best result is a skewed line that runs 19.9 feet SE of point “1” to 8.4
feet SE of point “3”.